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English Pages 432 [430] Year 2020
EXPLORING DATA ANALYSIS
Exploring Data Analysis The Computer Revolution In Statistics Edited by W. J. DIXON Department of Biomathematics University of California, Los Angeles
and W. L. NICHOLSON Battelle Pacific Northwest Laboratories and National Bureau of Standards
UNIVERSITY OF CALIFORNIA PRESS
Berkeley
Los Angeles
London
University of California Press Berkeley and Los Angeles, California University of California Press, Ltd. London, England ISBN: 0-520-02470-2 Library of Congress Catalog Card Number: 73-78549 Copyright © 1974 by The Regents of the University of California Printed in the United States of America
CONTENTS Prcfctcs
xviii
W. J. D i x o n Chapter 1
1
A D V A N C E D B R E A S T CANCER DATA J a m e s D i c k e y and Judy Walrath ( D i s c u s s i o n : R . M . E l a s h o f f , R. A. K r o n m a l and J. Tukey)
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Chapter 2 BLOC VOTING IN THE UNITED NATIONS J. A. Hartigan ( D i s c u s s i o n : L. A. J a e c k e l and R. M o o r e ) Chapter 3
109 U3
CRAWFORD HILL R A I N F A L L D A T A L o u i s A. J a e c k e l and John D. Gabbe ( D i s c u s s i o n : R. M o o r e , J. D i c k e y and J. Tukey) Chapter 4
177 181
EEG F R E Q U E N C Y DISTRIBUTION DATA M.R. Mickey ( D i s c u s s i o n : D. M a r t i n and J. Tukey) Chapter 5
222 229
ROBUSTNESS STUDY ANALYSIS (A F u r t h e r A n a l y s i s of the F i r s t P h a s e of the P r i n c e t o n R o b u s t n e s s Study; E x a m p l e s of L e s s Standard T w o - W a y Table A n a l y s i s ) John W. Tukey ( D i s c u s s i o n : J. Hartigan and H. L. L u c a s )
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294
Contents
Chapter 6
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ANALYZING A SERIES OF SOIL F E R T I L I T Y E X P E R I MENTS FOR PREDICTION F . B . Cady, R. L. A n d e r s o n and D. M. A l l e n ( D i s c u s s i o n : R. M. E l a s h o f f and J. Tukey) Chapter 7
325 331
EVOKED O F F R E S P O N S E TO AN AUDIO TONE D. C. M a r t i n and H. L. L u c a s ( D i s c u s s i o n : M. R. M i c k e y and J. Tukey) Chapter 8
355 365
THE USE OF DENSITY ESTIMATES B A S E D ON ORTHOGONAL EXPANSIONS R i c h a r d A. K r o n m a l and M i c h a e l T a r t e r (Discussion:
D . M . A l l e n and J. Tukey)
Citation Index
396 ¿ni
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LIST OF TABLES Chapter 1 1. The weighted likelihood ratio f o r equal nonfailure rate f o r adrenalectomy and hypophysectomy, subsample of mastectomy-negative cases. 2. The weighted likelihood ratio f o r equal nonfailure rate for adrenalectomy and hypophysectomy, subsample of mastectomy-negative premenopausal-when-first-seen patients. 3. P e r f o r m a n c e of discriminant analysis procedures. (Discussion: R. M. Elashoff) 1. 2. 3. 4. 5.
The prediction of P. Classification of cases by one multiple logistic. Type of operative treatment. Classification of cases by discriminant function. Ball smoothing classification.
Chapter 2 1. Vote counts in the General Assembly, 1968. 2. Selected contingency tables, 1968. 3. Clustering countries and issues: votes of 19 selected nations on 12 selected issues, General Assembly, 1968. 4. Amalgamation distances. 5. Tree diagram of cluster based on weighted average algorithm (12 variables). 6. Clusters of issues with distance = 100 (1 - ABS (correlation)). Based on 19 countries, amalgamating clusters in order of average distance between them. 7. Direct clustering of U.N. data. F r o m initial marginal t r e e s in tables 1 and 3. 8. Direct clustering of data on agreement with USSR. 9. Clusters of countries based on weighted algorithms. 10. Tree diagram of cluster based on weighted averages algorithm (50 variables). 11. Direct clustering of data on agreement with USA. 12. Direct clustering of U.N. votes.
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Chapter 3 1. E s t i m a t e d d r i f t f o r r a i n s t o r m s in 1967. 2. E s t i m a t e d d r i f t s f o r the quadrants of the g r i d for two r a i n s t o r m s in 1967. 3. R e s u l t s of fitting attenuation to r a i n r a t e . 4. Gauge n u m b e r s , n o m i n a l grid c o o r d i n a t e s and r a i n r a t e s . 5. S e l e c t e d t i m e - s m o o t h e d data, f i t t e d v a l u e s and r e s i d u a l s . 6. P a r a m e t e r s , d e s c r i p t o r s and other s t a t i s t i c s f r o m the i n d i vidual f i t s t o e q u a l l y s p a c e d s c a n s . 7. Standard e r r o r s and c o r r e l a t i o n s f o r the c o e f f i c i e n t s of s c a n 126. 8. P a r a m e t e r s and other s t a t i s t i c s f o r the t i m e - d e p e n d e n t f i t . 9. S o m e d e s c r i p t o r s of the t i m e - d e p e n d e n t f i t . 10. Mean r e s i d u a l s o v e r all 49 s c a n s in t a b l e 5. Chapter 4 1. E x p e r i m e n t s in f i r s t data s e t . 2. D i s c r i m i n a n t a n a l y s i s c l a s s i f i c a t i o n of l a s t e x p e r i m e n t on b a s i s of f i r s t and f i r s t on b a s i s of l a s t . 3. D i s c r i m i n a n t a n a l y s i s c l a s s i f i c a t i o n of one c a t on b a s i s of data f r o m two other c a t s . 4. R e s u l t s of fitting r e s p o n s e c u r v e to high d o s e p h y s o s t i g m i n e , cat No. 9. Chapter 5 (Discussion:
J. Hartigan)
1. T w o - w a y c l u s t e r i n g of data u s i n g 65 e s t i m a t o r s , t i o n s and the 10% e r r o r e s t i m a t e .
18 s i t u a -
Chapter 6 1. The independent v a r i a b l e s and the e s t i m a t e d p a r t i a l r e g r e s s i o n c o e f f i c i e n t s f o r the f u l l and r e d u c e d m o d e l s .
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Chapter 7 1 and 2. E l e c t r i c a l a c t i v i t y of t r e a t m e n t f r o m s t r i p c h a r t r e c o r d e r showing (A) off r e s p o n s e and (B) on r e s p o n s e . 3. A v e r a g e r e s p o n s e c u r v e with a two s t a n d a r d d e v i a t i o n conf i d e n c e i n t e r v a l (1000 Hz sine w a v e s t i m u l u s ) . 4 , 5 and 6. A v e r a g e e v o k e d r e s p o n s e plots showing s t a n d a r d d e v i a t i o n , two s t a n d a r d d e v i a t i o n , and two s t a n d a r d d e v i a t i o n of the m e a n e s t i m a t e s of the c o n f i d e n c e i n t e r v a l . 7. A v e r a g e evoked r e s p o n s e and two s t a n d a r d d e v i a t i o n c o n f i d e n c e i n t e r v a l with an e l e v e n t e r m t r u n c a t e d F o u r i e r a p p r o x i mation. (Discussion:
M. R . M i c k e y )
1. E v o k e d off and on r e s p o n s e to an audio tone a r r a n g e d a c c o r d ing to g r a d a t i o n of r e s p o n s e . Chapter 8 1. E s t i m a t e of d e n s i t y f o r s a m p l e of s i z e 200 f r o m N(0, 1). 2. E s t i m a t e of c u m u l a t i v e d i s t r i b u t i o n f o r s a m p l e of s i z e 200 f r o m N(0, 1). 3. E s t i m a t e of c o n t o u r s of p r o b a b i l i t y d e n s i t y f o r N(0, 1 , 0 , 1 , . 8). 4. D e n s i t y e s t i m a t e f o r r e s t i n g h e a r t r a t e f o r the patient g r o u p . 5. D e n s i t y e s t i m a t e f o r m a x i m u m h e a r t r a t e f o r the patient g r o u p . 6. E s t i m a t e of b i v a r i a t e d e n s i t y of m a x i m u m and r e s t i n g h e a r t r a t e in the patient g r o u p . 7. C o n t o u r s of the e s t i m a t e d p r o b a b i l i t y d e n s i t y f o r m a x i m u m and r e s t i n g h e a r t r a t e in the n o r m a l g r o u p . 8. C o n t o u r s of the e s t i m a t e d p r o b a b i l i t y d e n s i t y f o r s y s t o l i c and d i a s t o l i c blood p r e s s u r e f o r the patient g r o u p . 9. E s t i m a t e of c o n t o u r s f o r s a m p l e f r o m b i v a r i a t e n o r m a l with m e a n s , v a r i a n c e s , and c o r r e l a t i o n e q u a l to the v a l u e s of t h e s e f o r the d a t a shown in f i g u r e 8. 10. E s t i m a t e d c o n t o u r s of the p r o b a b i l i t y d e n s i t y f o r s y s t o l i c and d i a s t o l i c blood p r e s s u r e in the n o r m a l g r o u p . 11. S c a t t e r g r a m of s y s t o l i c v e r s u s d i a s t o l i c blood p r e s s u r e f o r the n o r m a l g r o u p .
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LIST OF FIGURES Chapter 1 1. F i r s t Look At Graphs (FLAG) output. 2. S u b s a m p l e H i s t o g r a m s On P l o t s (SHOP) output. 3. S m o o t h e d d i s c r i m i n a t i o n p r o b a b i l i t y e s t i m a t i o n output. Chapter 2 1. H i s t o g r a m s of counts of voting c a t e g o r i e s , o v e r 50 q u e s t i o n s . Chapter 3 1. The r a i n gauge n e t w o r k . 2. I s o m e t r i c plots of r a i n f a l l s (a) 7 / 2 8 A , and (b) 7 / 1 1 . 3. E x a m p l e s of the p a i r w i s e c o r r e l a t i o n s a s f u n c t i o n s of l a g , for s t o r m 7/25. 4. R e s u l t s of "pseudojackknife" for the r a i n f a l l of 7 / 2 5 . 5. D r i f t v e c t o r s for e a c h of the 42 5 x 5 s q u a r e s , a r r a n g e d a c cording to p o s i t i o n of the s q u a r e ( s t o r m 7 / 2 8 A ) . 6 . P r e d i c t e d v a l u e s of attenuation f r o m fitting the r a i n r a t e s of g a u g e s 9, 17 and 25 for r a i n f a l l 7 / 2 8 A . 7. P r e d i c t e d v e r s u s o b s e r v e d attenuation f r o m fit of r a i n r a t e s of g a u g e s 9, 17 and 25 to the attenuation for r a i n f a l l 7 / 2 8 A . 8. R e s i d u a l s of the f i t of r a i n r a t e s of g a u g e s 9, 17 and 25 to the attenuation for r a i n f a l l 7 / 2 8 A . 9. N o r m a l plot of r e s i d u a l s f r o m the fit to 49 t i m e s m o o t h e d scans. Chapter 4 1. P l o t f r o m p r o g r a m BMD07M showing c l u s t e r i n g of EEG a m p l i tude p r o f i l e s . 2. R e s p o n s e to high d o s e p h y s o s t i g m i n e , cat No. 9. 3. C h a r a c t e r i s t i c p r o f i l e s of high d o s e p h y s o s t i g m i n e , cat No, 9. 4. A v e r a g e s of c a n o n i c a l v a r i a b l e s for d o s e r e s p o n s e , and c h a r a c t e r i s t i c p r o f i l e s c o r r e s p o n d i n g to the c a n o n i c a l a x e s . 5. C a n o n i c a l v a r i a b l e s for d o s e r e s p o n s e .
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LIST OF EXHIBITS Chapter 4 1. Canonical c o r r e l a t i o n / m u l t i v a r i a t e r e g r e s s i o n analysis using p r o g r a m BMDX75. Chapter 5 1. The data as submitted - and later put in o r d e r . 2. Steps 1, 2 and 3 analyses. 3. Comparison of fit to 20 selected rows (20 selected e s t i m a t e s ) in the 8 x 65 and 8 x 20 of steps 1 and 2 for situation 2. 4. Comparison of " c o l " and " c o l * " for five situations. 5. Residuals f r o m r o w s - P L U S - c o l s fit to the 20 x 8 f r o m situation 16, as was and the signs after change. 6. P a r t s of the stages of analysis of the logs. 7. Comparison of the residuals after two fits of the f o r m common + row + col + r o w * • c o l * . 8. C l a s s i c a l r o w s - T I M E S - c o l u m n s fit to c o l * (eigen) effects. 9. Resistant r o w * * - T I M E S - c o l * * fit to " c o l " 8 - v e c t o r s for selected situations. 10. Intercept and residuals f r o m r e g r e s s i o n . 11. Step 7 analyses. 12. Latent values corresponding to r o w * - T I M E S - c o l * fits in steps 3 and 7. 13. The col* values found in step 7. 14. The row* values obtained in steps 3 and 7 for three f a r - f r o m Gaussian situations. 15. Step 8 analyses. 16. Step 9 analyses. 17. Two 8 - v e c t o r s plotted against (deviate)^. 18. Two m o r e 8 - v e c t o r s plotted against (deviate)^. 19. F i n a l calculation of estimate deficiencies. 20. Adjusted discrepancies (in . 0 0 1 ) for some selected e s t i m a t o r s for situation 22. 21. Adjusted discrepancies (in . 001) for selected e s t i m a t e s in situation 18.
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PREFACE
The g e n e s i s of this book w a s a c o n f e r e n c e on s t a t i s t i c a l c o m puting, o r g a n i z e d as a w o r k s h o p , to e x a m i n e the f r o n t i e r s of data a n a l y s i s b a s e d on c o m p u t e r u s e . It was held in the Health S c i e n c e s Computing F a c i l i t y (HSCF) at the U n i v e r s i t y of C a l i f o r nia at L o s A n g e l e s in S e p t e m b e r 1971. The o r i g i n a l i m p e t u s for s u c h a w o r k s h o p c a m e f r o m d i s c u s s i o n s with W e s l e y N i c h o l s o n during an international m e e t i n g in London s o m e y e a r s e a r l i e r . We w e r e d i s m a y e d at the c u r r e n t i v o r y tower t r e n d s in s t a t i s t i c s . M i m i c k i n g the m a t h e m a t i c i a n s , s t a t i s t i c i a n s w e r e i n c r e a s i n g l y avoiding the r e a l w o r l d of a p p l i c a t i o n , and w e r e p u r i f y i n g and s e p a r a t i n g the f i e l d f r o m o t h e r s c i e n c e s . The c o n f e r e n c e w a s planned as a c o u n t e r r e v o l u t i o n to that trend. The Health S c i e n c e s Computing F a c i l i t y p r o v i d e d an e x c e l l e n t p l a c e for the w o r k s h o p . The f a c i l i t y is d e d i c a t e d to s e r v i n g b i o m e d i c a l r e s e a r c h through r e s e a r c h in m a t h e m a t i c s , s t a t i s t i c s and c o m p u t e r s c i e n c e . It has an IBM 3 6 0 / 9 1 and n u m e r o u s t y p e w r i t e r , c h a r a c t e r s c o p e , and g r a p h i c s c o n s o l e s s e r v e d by a t i m e - s h a r i n g o p e r a t i n g s y s t e m . The s y s t e m s p e c i a l i z e s in i n t e r a c t i v e s t a t i s t i c a l t e c h n i q u e s and the p r o g r a m s to s e r v e t h e m . Of s p e c i a l i m p o r t a n c e to c o n f e r e n c e p a r t i c i p a n t s w a s the u s e of graphical statistical techniques. P a r t i c i p a n t s w e r e l i m i t e d to a s e l e c t group of p r a c t i c i n g data a n a l y s t s . The p a p e r s p r e s e n t e d r e a l p r o b l e m s and included a d i s c u s s i o n of the p h y s i c a l m e c h a n i s m s involved in g e n e r a t i n g data f o r the a n a l y s e s . With a r e a l p r o b l e m a s a f o c a l point, the a n a l y s e s p u r s u e d the needs of the p r o b l e m r a t h e r than s t r e s s i n g p a r t i c u l a r t e c h n i q u e s of s t a t i s t i c s . But any new t e c h n i q u e s u s e f u l
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f o r the a n a l y s e s w e r e e m p h a s i z e d , and the d e g r e e to which the d e r i v a t i o n and u s e of the t e c h n i q u e s w a s dependent on the c o m puter was s t r e s s e d . E a c h p a p e r w a s a v a i l a b l e to s e v e r a l c r i t i c s in a d v a n c e of the m e e t i n g . T h e i r c o m m e n t s a r e included in this v o l u m e a s well a s a d d i t i o n a l c o m m e n t s b y the a u t h o r s and other c r i t i c s that d e v e l o p e d during the s e s s i o n s . The c o n f e r e n c e r e v e a l e d m a n y c h a r a c t e r i s t i c s of a d a t a a n a l y s t a t work. In c o n t r a s t to the b i o l o g i s t who e x a m i n e s his data with the cons t r u c t s of his own f i e l d in m i n d , the data a n a l y s t e x a m i n e s the data f o r i t s a p p a r e n t s i m i l a r i t y to a v a r i e t y of s t a t i s t i c a l m o d e l s he h a s in m i n d , letting the r e s u l t s of s u c c e s s i v e a n a l y t i c a l a t t e m p t s guide the d i r e c t i o n he p u r s u e s (and r e f i n e s ) a s he p r o c e e d s . The s t a t i s t i c i a n ' s a p p r o a c h m i g h t b e d e s c r i b e d a s one in which he s t a t e s : "if we a s s u m e n o r m a l i t y , i n d e p e n d e n c e , and p e r h a p s other f u n d a m e n t a l s , then the r e s u l t s i n d i c a t e the v a l i d i t y of c e r t a i n s t a t e d h y p o t h e s e s with a s s o c i a t e d p r o b a b i l i t i e s . " In c o n t r a s t , the data a n a l y s t m a y u s e m a n y of the s a m e t e c h n i q u e s , but he will e x p l o r e ( a l s o with s t a t i s t i c a l t e c h n i q u e s ) the d e g r e e to which t h e s e a s s u m p t i o n s might b e a f f e c t i n g h i s c o n c l u s i o n s , and the c o n s e q u e n c e s to the a p p l i c a t i o n a l f i e l d of d e v i a t i o n s f r o m r e a l i t y in the a n a l y t i c a l a s s u m p t i o n s . The data a n a l y s t s e e m s to b e m o r e involved in e x p l o r a t i o n than in r e f i n e m e n t . He i s s l o w to m a k e a s s u m p t i o n s b e f o r e he e x a m i n e s the d a t a . He i s quite s a t i s f i e d if any a d v a n c e i s m a d e in the p r o b l e m a r e a independent of the s o p h i s t i c a t i o n of the a n a l y s i s , the g o o d n e s s of a g r e e m e n t of his m o d e l , or the p r e s e n t a b i l i t y of the s t a t i s t i c a l a n a l y s i s i t s e l f . He i s quite p r e p a r e d to find that one m i g h t a r r i v e at the s a m e c o n c l u s i o n u s i n g quite d i f f e r e n t r o u t e s and quite d i f f e r e n t t e c h n i q u e s . The data a n a l y s t i s a l m o s t s u r e to have a d e e p i n v o l v e m e n t in c o m p u t e r s s i n c e he r e q u i r e s computing power f o r h i s f r e e d o m to u s e a wide v a r i e t y of t e c h n i q u e s . T e c h n i q u e s and a n a l y s t s a r e not independent. They i n t e r a c t . One obtains a m a x i m u m r e s u l t f r o m i n t e r a c t i o n s r a t h e r than f r o m m a i n e f f e c t s . A p a r t i c u l a r p e r s o n who u s e s c e r t a i n xiv
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techniques m o r e powerfully than someone else m a y obtain b e t t e r a n s w e r s u s i n g t h o s e techniques than o t h e r s can. On the other h a n d , a n o t h e r p e r s o n m a y u s e h i s o w n t e c h n i q u e s a n d do e q u a l l y well, that i s , t h e r e is a n i n t e r a c t i o n in the p r o c e s s . Even when techniques a r e m a t h e m a t i c a l l y equivalent, different a n a l y s t s u s e t h e m i n d i f f e r e n t w a y s . O n e m a y t h i n k a n d do a n a l y s i s of v a r i a n c e , a n d a n o t h e r m a y t h i n k r e g r e s s i o n . They m a y b e doing the s a m e thing but their thought p r o c e s s e s and the w a y t h e y p r o c e e d t h r o u g h t h e a n a l y s i s of t h e p r o b l e m d i f f e r b e c a u s e of t h e w a y t h e y c o n c e p t u a l i z e a n a l y s i s of v a r i a n c e a n d r e g r e s s i o n ; although the l a n g u a g e m a y d i f f e r and e v e n c o m m u n i cation m a y b e difficult, the o v e r a l l a n a l y s e s m a y r e a l l y b e v e r y similar. B y t h e e n d of t h e c o n f e r e n c e i t w a s c l e a r t h a t t h e r e i s a h e a v y i n t e r a c t i o n b e t w e e n a n a l y s t s a n d s c i e n t i s t s i n o t h e r f i e l d s . In m o s t c a s e s , the a n a l y s t h a s b e c o m e v e r y involved with the s u b j e c t m a t t e r of t h e f i e l d ' s b a s i c t h e o r i e s a n d p r o b l e m s . The statistical r e s e a r c h for his data analysis is truly collaborative— i n m a n y c a s e s h e e n l i s t s t h e c o o p e r a t i o n of o t h e r s t a t i s t i c i a n s a s w e l l . T h e s t a t i s t i c a l a n a l y s i s is not s e p a r a t e d and p u r s u e d f o r its m a t h e m a t i c a l e l e g a n c e , r a t h e r it is o r i e n t e d t o w a r d the n e e d s of t h e p r o b l e m . P e r h a p s this " t e a m " w o r k and c o o p e r a t i o n is the m o s t i m p o r t a n t a n d f a r - r e a c h i n g r e v e l a t i o n of t h e c o n f e r e n c e . A s h o r t d e f i n i t i o n of d a t a a n a l y s i s w a s p r o p o s e d a t t h e c o n f e r e n c e : D a t a a n a l y s i s i s t h e a p p l i c a t i o n of o n e o r m o r e t e c h n i q u e s t o a s e t of d a t a s t e e r e d b y t h e p r o b l e m . C o m p u t e r f a c i l i t i e s a t H S C F w e r e a v a i l a b l e to p a r t i c i p a n t s b e f o r e and during the c o n f e r e n c e , and a UCLA "buddy" was a s s i g n e d to e a c h p a r t i c i p a n t to h e l p in a n y w a y n e c e s s a r y . Data p r e s e n t e d at the c o n f e r e n c e is a v a i l a b l e f r o m H S C F in m a c h i n e r e a d a b l e f o r m . A d a t a s e t d e s c r i p t i o n c o n t a i n i n g a t l e a s t a p a r t i a l l i s t i n g of t h e d a t a f r o m e a c h p a p e r is g i v e n i n t h i s b o o k . T h e c o m p u t a t i o n a l s u p p o r t w a s m a d e p o s s i b l e b y g r a n t R R - 3 of t h e B i o t e c h n o l o g y R e s o u r c e s B r a n c h , D i v i s i o n of R e s e a r c h a n d
xv
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R e s o u r c e s of the National Institutes of Health. The c o n f e r e n c e i t s e l f w a s supported b y grant G J - 2 9 8 4 4 f r o m the National S c i e n c e Foundation. A c k n o w l e d g e m e n t s a r e due s e v e r a l m e m b e r s of m y staff for their h e l p with the c o n f e r e n c e and in p r e p a r i n g m a t e r i a l f o r this book. Ed Chen, D o l o r e s A d a m s and E l l e n S o m m e r s a s s i s t e d in p r e p a r a t i o n s f o r and during the c o n f e r e n c e . E l l e n S o m m e r s p r e p a r e d and edited the a s s o c i a t e d data s e t s . Lyda B o y e r e d i t e d , and B e t s y P o t t e r typed the m a n u s c r i p t s . Much of the work o r o r g a n i z a t i o n of the c o n f e r e n c e i t s e l f and w o r k i n g with the authors on the p r e p a r a t i o n of t h e i r m a n u s c r i p t s w a s done by W e s l e y N i c h o l s o n .
W. J. Dixon
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CHAPTER
1
A D V A N C E D B R E A S T CANCER DATA
JAMES DICKEY S t a t i s t i c s D e p a r t m e n t , State U n i v e r s i t y of New York at B u f f a l o and JUDY WALRATH D e p a r t m e n t of E p i d e m i o l o g y and P u b l i c Health, Yale U n i v e r s i t y
The m a j o r i t y of m e d i c a l d a t a - a n a l y s i s p r o b l e m s a r i s e f r o m a p h y s i c i a n ' s hope that h i s r e c o r d s of p a s t c a s e s w i l l y i e l d u s e f u l i n f o r m a t i o n . The r e a l p r o b l e m s a r e m a t h e m a t i c a l l y v a g u e , but tangible: What l e s s o n s a r e to be l e a r n e d f r o m past e x p e r i e n c e f o r future c l i n i c a l p r a c t i c e ? What patient subpopulations have d i s t i n c t i v e b e h a v i o r p a t t e r n s ? What t r e a t m e n t s should be u s e d in what kinds of c a s e s ? In the language of John Tukey (1962, 1970), t h e s e a r e p r o b l e m s of " e x p l o r a t o r y data a n a l y s i s " — p r o b l e m s of how to Find I n t e r e s t i n g R e p o r t a b l e E f f e c t s (FIRE). F I R E p r o b l e m s , h o w e v e r , are not the subject of the bulk of s t a t i s t i c a l t h e o r y , w h i c h i s d e v i s e d f o r A f t e r The R e v e l a t i o n O r d e r l y P i c k l i n g of H Y p o t h e s e s (ATROPHY), and to Guard A g a i n s t S i l l y S e l e c t i o n E f f e c t s by D e f i n i t i o n (GASSED). R e s e a r c h for this study w a s supported by NIGMS-NIH Grant GM 16557.
1
Dickey and Walrath
L i n e a r d i s c r i m i n a t i o n p r o c e d u r e s have not b e e n v e r y p r o d u c t i v e i n r e a l m e d i c a l p r o b l e m s ( R a d h a k r i s h n a , 1964). E v e n t h e F I R E p r o b l e m - m o t i v a t e d s t e p w i s e l i n e a r p r o c e d u r e s ( r e g r e s s i o n and d i s c r i m i n a t i o n ) deliver l i n e a r functions that tend to be a l m o s t m e a n i n g l e s s a s f i n a l a n s w e r s t o p h y s i c i a n s and s t a t i s t i c i a n s a l i k e , e s p e c i a l l y l i n e a r f u n c t i o n s of t h r e e o r m o r e v a r i a b l e s . They m a y , h o w e v e r , be u s e f u l in pointing out the f e w i m p o r t a n t v a r i ables. In t h i s p a p e r we s t r i v e t o c o n c e n t r a t e on F I R E p r o b l e m s of c l i n i c a l - e x p e r i e n c e d a t a , w i t h t h e a i m of c o n t r i b u t i n g t o a g e n e r a l s y s t e m a t i c a p p r o a c h i n v o l v i n g t h e u s e of c o m p u t e r p r o g r a m s a s s t e p s i n a n a n a l y t i c s e q u e n c e . We d i s c u s s e x p l o r a t o r y d a t a a n a l y s i s f o r a n i m p o r t a n t c l a s s of p r o b l e m s — t h e p r e d i c t i o n of a dichotomized treatment-response variable. P r o f . W i l f r i d J . D i x o n ' s ( 1 9 6 9 , 1970) B M D b i o m e d i c a l c o m p u t e r p r o g r a m s are widely used for practical data analysis. Contribut i o n s to a s y s t e m a t i z e d a p p r o a c h , i n s p i r e d b y t h e B M D p r o g r a m s , a r e put f o r t h h e r e , t o g e t h e r w i t h a f e w r o u g h p r e d e c e s s o r F O R T R A N l a n g u a g e p r o g r a m s , and p r o g r a m s n o t y e t a v a i l a b l e . In the following s e c t i o n we i n t r o d u c e , as c o n c r e t e m o t i v a t i o n , t h e w e l l - s t u d i e d ( A r m i t a g e e t a l , 1969) a d v a n c e d b r e a s t c a n c e r d a t a a n a l y s i s , a n d t h e c l i n i c a l - d e c i s i o n p r o b l e m of B u l b r o o k e t a l ( I 9 6 0 ) , and A t k i n s e t a l (1968). E a c h of t h e r e m a i n i n g s e c t i o n s d e s c r i b e s a t y p e of c o m p u t e r p r o g r a m : •
F i r s t L o o k At G r a p h s
•
S u b s a m p l e H i s t o g r a m s O r P l o t s (SHOP);
•
S h o p In F u l l T o t a l i t y ( S I F T ) ; and
•
a d i s c u s s i o n of d i s c r i m i n a n t a n a l y s i s p e r s e , w i t h an e m p h a s i s on r e c e n t n o n p a r a m e t r i c p r o c e d u r e s .
2
(FLAG);
Advanced Breast Cancer
A DATA ANALYSIS PROBLEM The t y p i c a l m e d i c a l d a t a s e t f e a t u r e s a f e w (1 - 10) r e s p o n s e v a r i a b l e s and m a n y (10 - 100) m i x e d - t y p e ( d i c h o t o m o u s t o p r a c t i c a l l y c o n t i n u o u s ) p r e d i c t o r v a r i a b l e s , f o r a p r e c i o u s f e w (10 1000) o b s e r v e d c a s e s . M i s s i n g v a l u e s abound. The d e f i n i t i o n s of i n d i v i d u a l v a r i a b l e s a r e a m b i g u o u s and i l l - c o n c e i v e d . The data e m b o d y h i s t o r i e s of u n d i s c i p l i n e d c l e r k s ' m i s u n d e r s t a n d i n g s . In s h o r t , the s t a t i s t i c s t e a c h e r ' s n i g h t m a r e : i m p e r f e c t d a t a and vague problems. We c o n s i d e r h e r e a d e c i s i o n p r o b l e m in the m a n a g e m e n t of a d v a n c e d b r e a s t c a n c e r , and a r e l a t e d data s e t f r o m Guy's H o s p i t a l , L o n d o n (Atkins et a l , 1968), u n u s u a l f o r the p a i n s taking c a r e w i t h w h i c h it w a s c o l l e c t e d . T h i s c o n c r e t e d a t a a n a l y s i s p r o b l e m i s put f o r t h a s r e p r e s e n t a t i v e of m a n y in b e i n g s u i t e d to a g e n e r a l s y s t e m a t i c a p p r o a c h . T w o h u n d r e d and t e n a d v a n c e d b r e a s t c a n c e r p a t i e n t s w e r e i n c l u d e d in the s t u d y . A p p r o x i m a t e l y t w o - t h i r d s ( 1 3 9 / 2 1 0 ) of t h e m had u n d e r g o n e a t t e m p t e d c u r e by r a d i c a l ( 1 1 6 / 2 1 0 ) or s i m p l e ( 2 3 / 2 1 0 ) m a s t e c t o m y , and t h e n a y e a r o r s o l a t e r had a r e c u r r e n c e of t u m o r g r o w t h l o c a l l y o r at a d i s t a n t s i t e . The o t h e r o n e - t h i r d ( 7 1 / 2 1 0 ) had b e e n f i r s t d i a g n o s e d as a l r e a d y a d v a n c e d . Threef i f t h s ( 1 3 2 / 2 1 0 ) b e g a n the p a l l i a t i v e s t a g e of t h e i r t r e a t m e n t w i t h the a d m i n i s t r a t i o n of h o r m o n e s , w h i c h w e r e u s e f u l in s o m e c a s e s ( 1 7 / 1 3 2 ) f o r up t o one y e a r in c o n t r o l l i n g t u m o r g r o w t h . T h e n it w a s a q u e s t i o n of w h e t h e r or not s u r g e r y s h o u l d b e u s e d to a l t e r the h o r m o n a l e n v i r o n m e n t of the t u m o r s . If s o , w h i c h of two o p e r a t i o n s s h o u l d b e p e r f o r m e d : b i l a t e r a l a d r e n a l e c t o m y w i t h o o p h o r e c t o m y ( r e m o v a l of a l l a d r e n a l s and o v a r i e s ) , or h y p o p h y s e c t o m y ( r e m o v a l of p i t u i t a r y ) . E a c h p a t i e n t u n d e r w e n t a n o p e r a t i o n , about half e a c h kind ( 1 1 5 / 2 1 0 , 9 5 / 2 1 0 ) . F o r o n e - q u a r t e r of the p a t i e n t s ( 5 4 / 2 1 0 ) , the s u r g e r y w a s s u c c e s s f u l ( c o m p l e t e r e m i s s i o n of s y m p t o m s f o r o v e r s i x m o n t h s ) ; f o r another o n e - q u a r t e r (53/210), intermediate r e s u l t s (partial r e m i s s i o n ) ; and f o r the o t h e r half ( 1 0 3 / 2 1 0 ) , f a i l u r e (no i m p r o v e ment). 3
Dickey and Walrath
B o t h s u r g i c a l p r o c e d u r e s a r e r a d i c a l a t t e m p t s to p r o l o n g l i f e . H y p o p h y s e c t o m y i s a m o r e i n v o l v e d and d a n g e r o u s o p e r a t i o n , b u t i t s w h o l e - s a m p l e r e m i s s i o n p e r c e n t a g e s ( 2 8 / 9 5 and 2 4 / 9 5 ) w e r e e s s e n t i a l l y t h e s a m e a s t h o s e f o r a d r e n a l e c t o m y ( 2 6 / 1 1 5 and 29/115). N a t u r a l suggestions for v a r i a b l e s r e l a t e d to s u r g i c a l s u c c e s s include: 1.
m e a s u r e s of t u m o r g r o w t h r a t e a) age of p a t i e n t b) e x t e n t of d i s e a s e at m a s t e c t o m y c) t i m e f r o m m a s t e c t o m y t o r e c u r r e n c e ;
2.
tumor histology;
3.
menopausal status;
4.
h i s t o r y of m a s t e c t o m y ;
5.
s y s t e m i c (hence u r i n a r y ) h o r m o n e l e v e l s .
and
In I 9 6 0 , D r . R . D. B u l b r o o k and h i s c o i n v e s t i g a t o r s at G u y ' s H o s p i t a l d e v e l o p e d a l i n e a r d i s c r i m i n a n t f u n c t i o n of two 2 4 - h o u r u r i n a r y - s t e r o i d l e v e l s , a e t i o c h o l a n o l o n e (E) and 1 7 - h y d r o x y c o r t i c o s t e r o i d (17 OHCS), 80 - 80(17 OHCS) + E ,
(1)
p o s i t i v e v a l u e s of w h i c h t e n d to p r e d i c t f a v o r a b l e r e s p o n s e t o s u r g e r y . A f t e r f u r t h e r p r o s p e c t i v e s t u d i e s , A t k i n s e t a l (1968) r e p o r t e d " t h e d i s c r i m i n a n t f u n c t i o n b y i t s e l f p r o v i d e s an e f f i c i e n t g u i d e to r e s p o n s e t o h y p o p h y s e c t o m y b u t d o e s n o t do s o f o r a d r e n a l e c t o m y in t h i s s e r i e s . " T h e y a l s o f o u n d s m a l l e f f e c t s f o r t h e f a c t o r s 1. c), 3. , and 4. a b o v e . A r m i t a g e e t a l (1969) c a r r i e d out e x t e n s i v e F I R E - l i k e a n a l y s e s of t h e s e s a m e d a t a . F i r s t , e a c h of t h r e e r e s p o n s e v a r i a b l e s w a s
4
Advanced Breast Cancer
d i c h o t o m i z e d and fit by H i l l s ' (1967) s t e p w i s e s ample - s plitting d i s c r i m i n a t i o n p r o c e d u r e for d i c h o t o m i z e d p r e d i c t o r v a r i a b l e s . Then they p e r f o r m e d s p e c i a l a n a l y s e s , e a c h suited to e a c h o r i g i nal r e s p o n s e v a r i a b l e . The r e s p o n s e , a c l i n i c a l a s s e s s m e n t of s u c c e s s (as s u c c e s s , i n t e r m e d i a t e , and f a i l u r e , defined above), w a s d i c h o t o m i z e d into nonfailure and f a i l u r e , and then r e l a t e d to v a r i o u s s e t s of p r e dictor v a r i a b l e s . Our d i s c u s s i o n i s r e s t r i c t e d to this c h o i c e of a d i c h o t o m o u s r e s p o n s e v a r i a b l e and to d i c h o t o m i z e d r e s p o n s e s in g e n e r a l , thus n e g l e c t i n g other important d e v e l o p m e n t s of m e t h o dology, for e x a m p l e , s u r v i v a l - t i m e data. At the s u g g e s t i o n , and through the k i n d n e s s , of P r o f . M a r v i n Z e l e n , a c a r d copy of the A r m i t a g e et al (1969) data w a s obtained f r o m John Copas, and a s l i g h t l y updated v e r s i o n of the o r i g i n a l patient r e c o r d s (including 16 new c a s e s ) f r o m Dr. R. D. Bulbrook. The updated r e c o r d s of all 210 c a s e s are on f i l e at HSCF under the t i t l e "Advanced B r e a s t Cancer Data (J. Dickey). " A c o m p l e t e l i s t i n g of the c a n c e r data in card i m a g e f o r m i s g i v e n in the Data Set D e s c r i p t i o n at the end of this c h a p t e r . This i n c l u d e s a d e s c r i p t i o n of the 50 v a r i a b l e s a s s o c i a t e d with e a c h patient, and, p a r e n t h e t i c a l l y , single word a c r o n y m s which identify v a r i a b l e s . FIRST LOOK AT GRAPHS (FLAG) N e w l y punched data w i l l , with high probability, contain m i s t a k e n v a l u e s appearing as 1.
o v e r p u n c h e s and i l l e g a l c h a r a c t e r s ;
2.
data-to-format mismatches;
3.
n o n s e n s e v a l u e s of a v a r i a b l e a) o f f - r a n g e n u m e r i c v a l u e s b) m e a n i n g l e s s m u l t i p l e - c h o i c e v a l u e s ;
4.
n o n s e n s e c o m b i n a t i o n s of v a r i a b l e v a l u e s , autopsy date p r e c e d i n g date of death; 5
e.g.,
Dickey and Walrath
5.
multivariate outliers;
and
6.
undetectable-per-se mistaken values.
C o m p u t e r p r o g r a m - p r o c e s s i n g s y s t e m s t e n d to a b o r t r u n s when d a t a input c o n t a i n s m i s t a k e s of t y p e s 1 and data-analysis programs abort or deliver unacceptable f r o m input m i s t a k e s of type 3, and l e s s c o m m o n l y , of
program 2. Many output type 4 .
One of the f u n c t i o n s of o u r c o m p u t e r p r o g r a m , F L A G ( G o l d m a n e t a l , 1 9 7 1 ) i s to d e t e c t , and i d e n t i f y b y f l a g g e d output, m i s t a k e n d a t a v a l u e s of t y p e s 1 - 4 without a b o r t i n g o r o t h e r w i s e d i s r u p t i n g p r o g r a m f u n c t i o n s . T h e p r o g r a m m i n g of m i s t a k e s 1 and 2 i s c o n c e i v e d a s i s o l a t e d in s y s t e m - s p e c i f i c s u b p r o g r a m s . T h e p r o g r a m h a s an option f o r a o n e - l i n e - p e r - c a r d p r i n t e r l i s t i n g of the d a t a , a s s u m i n g input i s in s t a n d a r d c a s e w i s e f o r m with a c o n s t a n t n u m b e r of c a r d s p e r c a s e . O p t i o n a l l y , s p e c i a l l i n e s on e a c h page should be u s e d to i n d i c a t e the a c t u a l c a r d c o l u m n s of each data v a r i a b l e . T h e p r i n c i p a l f u n c t i o n of F L A G i s to d e l i v e r c o a r s e p a r a l l e l p l o t s of t h e v a r i a b l e s a g a i n s t s e q u e n t i a l c a s e n u m b e r s with f l a g s f o r m i s s i n g v a l u e s and m i s t a k e s 1 - 4 . F i g u r e 1 d e p i c t s s u c h output for our f i r s t nine p r e d i c t o r v a r i a b l e s . E a c h print column c o r r e sponds to a v a l u e o r g r o u p i n g i n t e r v a l f o r a v a r i a b l e . T h e p r i n t c o l u m n s should be h e a d e d with the a c t u a l d a t a v a l u e s t o o , r a t h e r than as p r e s e n t l y k e y e d . And s p e c i a l p r i n t c o l u m n s should b e u s e d f o r u s e r - p r o g r a m m e d n o n s e n s e c h e c k s ( m i s t a k e of type 4 ) . P r o f . C h e s t e r I . B l i s s s t a t e d in c o n v e r s a t i o n with one of the a u t h o r s t h a t d a t a a n a l y s i s b y d e s k c a l c u l a t o r h a s the a d v a n t a g e of f o r c i n g the a n a l y s t into i n t i m a t e f a m i l i a r i t y with the a c t u a l d a t a v a l u e s . We hope F L A G o f f e r s s o m e of the s a m e a d v a n t a g e s . R u n s , t r e n d s , and m a n y o t h e r u n i v a r i a t e p a t t e r n s a r e e a s y to r e c o g n i z e . T o a l e s s e r e x t e n t , m u l t i v a r i a t e p a t t e r n s c a n be recognized, for example, simultaneously missing values.
6
Advanced Breast Cancer
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18
Advanced Breast Cancer
p ( D | H) = E ~
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p(D|7T),
TT | -H-
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~ ~
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p(D ~ | H ) = E 7TI, -rlp ( D~| 7 T ~), (D not r a n d o m , n o r c o n d i t i o n e d on, in the e x p e c t a t i o n s ) , w h e r e £ r e p r e s e n t s the p a r a m e t e r s of the s t a t i s t i c a l m o d e l s P(D|TT). The w e i g h t e d l i k e l i h o o d r a t i o i s i n t e r p r e t e d a s the r a t i o of the p o s t e r i o r o d d s f o r H to the p r i o r o d d s , no m a t t e r what the (finite n o n z e r o ) p r i o r o d d s , P ( H | D ) / P ( H | D) = L
(H, H) •
P(H)/P(H).
(7)
H e n c e the w e i g h t e d l i k e l i h o o d r a t i o r e p o r t s the i n f e r e n t i a l content of the d a t a D f o r H. The w e i g h t e d l i k e l i h o o d r a t i o i s a f u n c t i o n of the p r i o r d i s t r i b u t i o n , and s o s h o u l d be r e p o r t e d f o r s e v e r a l c h o i c e s of p r i o r d i s t r i b u t i o n u n d e r H. The p r i o r d i s t r i b u t i o n s u n d e r the s i n g u l a r s u b s e t H a r e a s s u m e d to b e i n d u c e d a s the l i m i t of c o n d i t i o n a l d i s t r i b u t i o n s u n d e r H ( S a v a g e ' s d e n s i t y p r i n c i p l e , D i c k e y and L i e n t z , 1970). The w e i g h t e d l i k e l i h o o d r a t i o s a r e g i v e n h e r e f o r t h r e e c o n t e x t s : 1.
2 x 2 4 - c e l l m u l t i n o m i a l m o d e l , h y p o t h e s i s of i n d e p e n d e n c e (Gunel and D i c k e y , 1972). D = (n
,n
,n
), n . . > 0, i, j = 1 , 2 , n
= N fixed.
D IT ~ M u l t i n o m i a l (TT, N). P(tt| H) = [ n . . ^ . - 1 ] r ( 4 a ) / [ r ( a ) l 4 , ~ li n i j iJ flattening constant
a. 19
a > 0, D i r i c h l e t with
Dickey and Walrath
[n.r(n. + 2 a - l ) ] [ n . r ( n L
(H,H)
1
=
[r(a)]4[r(4a-2)]2 [r(2a-l)]
2.
4
r
+ 4a)
—
[nijr(nij+a)][r(n
.
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i
+4a-2)]
.
(
(4a)
T w o - b i n o m i a l m o d e l , h y p o t h e s i s of e q u a l r a t e s ( D i c k e y a n d Lientz, 1970). For i = 1,2, D. = n . , , n . f i x e d , I ll 1»
£= ( W -
V W
D. TT ~ B i n o m i a l (TT.,n. ), i n d e p e n d e n t f o r i = 1 , 2 . l ~ l i * P(7T I H) = TT^"1 (1 - T T . ) a " 1 / B ( a , a ) , a > 0, b e t a w i t h p a r a m e t e r s (a, a ) , i n d e p e n d e n t f o r i = 1 , 2 . H:
7T1 = TT2.
D =
(DlfD2). [n.r(n
L
d
( H , H ) =
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.+ 2 a - l ) ] [ n i r ( n .
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[n..r(n..+a)]r(n
i]
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' 2
(
Advanced Breast Cancer
3.
O n e - b i n o m i a l m o d e l , h y p o t h e s i s of a f i x e d r a t e ( J e f f r e y s , 1961, p. 256). D = s , f i x e d n. D 17T ~ B i n o m i a l (tt , n). P ( t t | h ) = 7T a_1 (1 - 7 T ) a _ : l / B ( a , a), a >
0, b e t a w i t h
p a r a m e t e r s (a, a ) . H : TT = 7T ri r
/«
tt
H
fixed,
„
r(n+2a)
[r(a)]
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S e e a l s o Good (1950, 1955) f o r s i m i l a r B a y e s f a c t o r s . T a b l e 1 r e p o r t s the e v i d e n c e f o r an o p e r a t i o n e f f e c t in the m a s t e c t o m y - n e g a t i v e subs a m p l e . Note that p r i o r odds for n o e f f e c t a r e d i m i n i s h e d b y t h e d a t a t o p o s t e r i o r o d d s of a b o u t 1% of t h e p r i o r o d d s and t h a t t h e s t r o n g e r e v i d e n c e f o r t h i s c h a n g e c o m e s f r o m the a d r e n a l e c t o m y c a s e s . Table 2 r e p o r t s the e v i d e n c e i n t h e m a s t e c t o m y - n e g a t i v e and p r e m e n o p a u s a l - w h e n - f i r s t s e e n s u b s u b s a m p l e ; h e r e t h e d i m i n u t i o n of o d d s i s t o a b o u t 3%, which is w e a k e r evidence f o r the analogous i n f e r e n c e . The w e a k e n i n g a p p e a r s t o f o l l o w f r o m t h e r e d u c t i o n in s a m p l e s i z e . T h e r e p o r t i n g of t h e s e s u b s a m p l e w e i g h t e d l i k e l i h o o d r a t i o s i s l e s s v u l n e r a b l e to s p u r i o u s s e l e c t i o n e f f e c t s f r o m the m u l t i p l i c i t y of c a n d i d a t e s u b s a m p l e s t h a n i s t h e u s e of t h e u s u a l s u b s a m p l e P v a l u e s . E a c h p r i o r (and h e n c e p o s t e r i o r ) o d d s r a t i o f o r n o - r e a l e f f e c t in a s u b p o p u l a t i o n t e n d s t o b e g r e a t e r t h e m o r e t h e n u m b e r of s u c h s u b p o p u l a t i o n s u n d e r c o n s i d e r a t i o n ; h e n c e t h e m o r e e x t r e m e m u s t b e e a c h w e i g h t e d l i k e l i h o o d r a t i o to a t t r a c t a t t e n tion.
21
Dickey and Walrath
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BEST
20
2027 2033 2107 2243 2330 2254 2 332 2453 2358 2417 2520 2457 2349 2447 2319 2358 2574 2603 2608 2502
2304 2323 2461 2598 2684 2752 2812 2653 2846 2864 2683 2876 2886 2943 2926 2953 3008 3046 3066 3045
ruATIQN
2
DATA
STEP 2
FOR
STEP
2057 2068 2153 228 1 2372 2310 2377 2499 2416 2470 2556 2509 2418 2525 2396 2432 2624 2646 2661 2570
2
21 5C 2165 2276 24 02 2504 2473 2525 2645 2571 2631 2684 2654 26 C 3 27C9 2599 263C 2779 2786 2816 2770
ROWS'- P L U S - C O L U M N S
ANALYSIS
2923
-50
-498
-450
-283
-586 -562 -413 -282 -191 -127 -88 -18 -11 -4 5 11 27 63 79 108 136 176 199 195
19 14 3 9 4 8 29 0 -13 -3 6 -6 -12 8 -24 -26 0 -2 -4 -22
190 172 97 102 98 -41 -2 48 -53 -1 91 22 -101 -39 -183 -173 14 2 -14 -116
172 159 95 92 92 -33 -5 46 -43 2 79 26 -80 -9 -154 -147 16 - 1 -9 -96
98 89 51 46 57 -37 -24 25 - 55 -3 40 4 -62 7 -118 -116 4 - 29 -21 -64
rUATION
2
STEP
3
2369 23 S 7 2557 2683 2779 2838 2865 2956 2976 2981 2973 2988 3C14 3055 3078 31C9 3110 3140 3178 32C3
ROWS - T I M E S - C O L U M N S A N A L Y S I S
OF
31C4 3196 3364 36 15 35 75 4359 4555 39 37 4 159 40 19 3931 3950 4336 42 30 4528 4507 4230 4445 4332 4332
BEST 20
332
571
1190
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- 2 06 -175 -108 -89 -134 63 61 -68 114 10 -91 -14 131 -25 256 233 -4 38 5 107
-422 -453 -335 -215 -346 373 531 -157 57 -89 -186 -174 195 53 3 36 286 -19 155 20 23
-195
- 3 50
- 8 16
-10 11 19 1 9 -52 -87 0 51 29 -5 29 20 -15 58 61 0 -13 7 59
-11 19 26 8 8 -61 -100 -1 65 33 -4 42 33 -41 78 75 5 -13 -5 57
30 0 -21 11 -15 82 154 -2 -55 -35 16 -42 -3 3 16 -77 -8 2 3 34
OF
20
281
250
169
-21
385 386 266 193 281 -246 -319 132 -95 46 173 112 -194 -31 -350 -312 19 -101 -20 -98
8 3 -3 3 -3 15 38 -3 -10 -4 1 -9 -6 9 -14 -17 0 0 -3 -19
34 16 -9 24 -15 59 127 -4 -14 -20 21 -22 -22 -26 -40 -46 7 44 -5 -76
33 20 0 22 -8 56 110 - 1 -8 -13 17 -13 -9 2 -27 -34 9 34 - 1 -61
4 -4 - 13 -1 - 11 22 53 -7 -32 -14 -1 -22 -15 14 - 32 - 40 0 -4
-6 -1 5 -2 5 - 15 -29 4 11 12 0 6 7 16 14 17 0 -13 4 30
240
2701 2755 7971 31 22 3168 3429 3466 34C7 3596 3500 3407 3490 3652 3531 3829 3835 3625 3709 3698 3797
51
1441
- 16 -40
2549 2594 27e5 291 9 2993 3144 31 6 9 3199 3322 3267 32C6 32 64 3358 3312 3492 3513 3386 3447 3467 3538
BEST
20
-4 -93
Robustness Study Analysis
EXHIBIT
2,
C ONT I NU EC
SITUATION
16
STEP
1
ROWS-PLUS-COLUMNS
2 549
17
-417
2 8 0 4 5 53 25222 4489
7192863 7263 1240
-2775004
17
-652
-631
-21 -32
172 232
157
950 -161 -7 -164 366
-20659 -3340
-16
193
8 -4 485 8430996 -17
-219 117 -966 -1543335 149
5 183 49
273 -4004 -2090 -1119 -245
108 -202
69 5 -7 -30
857406 1966
-70 1448 1566421
-390 -1B42943 - 19984 -3185
218 170 -226 100
ANALYSIS
- 15
350
7 02
2063
1516319 - 7262 -see
1516320 24813 889
7192179
-16814 -25S2
59031 4241
7190816 1337 27 1R699
-5C6
- 17
632
1447
3976
22 33
-138 -2C5
-1499
17
- 1 59
-268 -441 -2 99
-288 -174C2C7
1 16 169 126 - 1 91 69
-
-7 5 -445
-953 -875681 132 240
-846 -e01847 97
-657647 17
172
-5
-2877 -1615
-182 -48
0 206
-3777 -2013 -1078 -243 -10 190
-49453 172
- 8 5 59 22 -925
-840161 -890
1010
109
-511
241 -51
57 0 -81
-105 78 -452 -67
-125 -486 6288 3314 1841 518
1411 497 508
1110 249
2403 629
7620 1445
0 30
21 -172
149 -364
188 -1306
49453 -172
1779537 944
2333160 2991
2981885
-714
-503
-421
-1C9
-113 66 -430
-113 35 -285
-56 -20
516 65 -104
1600 394
-67 -73
163
301 24 154
- 13 - 16 222 40 125
- 16 146 -638191
e2 132 135
-43 -35
-335 -520 -609
-38 -28 -25
123 213 276
117 200 256
98 154 195
-456 -302 -417
-2 3 -33 -27
197 104 277
184 102 257
141
38 28 25 24
85 196
34 28
-534 8566 -475
-21 3260 -20
343 -7670 376
-7473 351
234 -6760 261
22 -3021
827
50
-39 5
-39 5
-335
-461
316 71 418 418 42
286 48 391 391
219
-315 -2548 -2548 -307
-23 0 -16 -16 0
-374
0
-507 1533
7 364
188 206 -590
-70
-33 -29 -18
-73 334 19
314
1 2 89
- 2 63 - 3 93 -491 - 3 42
- 1 1 15 -1323 -1553 -1137 -929 -1604
- 2 84
-215 -493 -581
3022 -316
12333 -644
541
-1777 51908 - 1 9 39
1360
2514 -1691
-210 -218
- 5 64 - 2 67 -701 -701 - 2 57 -404
- 2 75 -249 -349
1306 -2062 -2062 271
-408 2892 0
1 72 199
30 18
-210 -251 266
-442 - 5 02 863
1931
-51
-83
- 5 51
-63 15
142 77
2252 192
15 -53
-258
5 00 231 - 5 09
113 1C49
5 03 2815
-186
32
0
-23
81
68
51
275 -89
64 -14
-257 -42
-253 -46
-215 587
-15 54
315 -184
280 -189
-216 - 39 2C5
1589 21489
272 16431
70345
26022 8724433
-3460 -373 368 8 39
-143 -1617 -8 81 -1229
-228 4 78 0
0
-61416 -878451 -3313 -364 334
-172 -99 -235
-205
323 -543 -114 -331
-6 -363 29
222 266 -214
-89 1 -18610
23 -1C6
-18 -151 -114 - 1 88 -239
1658 0 592
-104 782 300
138 -563 3
0
-900 -19150 -64205 - 1 2 5 1956
20 -49
230 -272
7 3 14 4292
130
289 0
33 173 19 1 -619
17 62 35
443 219
28355 12357
16 16 -60 0
237 287 -227
-22 44
16432 8428949 - 7 26 - 6 94
- 68 -4
-863 -194
-73 -419
1366 53
-211
-923 1527 - 3 19
-465 6229 3793
-356
1272981 4783 662 -670 1839
446 657648 -119
768 -124 3772 6262058
-256 2725 1725
14 17 -17
-294 -502 574
3C2 -97
-966
-162 -791 -15478 -53291 - 8 362 2 5
- 31 24
-271 -3492 -26021 -60e768
-2718 -3C9
-978 - 53
2 52 454
23 -44
731
241
-349
3493 97181 608769 978 3 68 -3C6 -522
12661 236016 5934445 5049 980 - 6 28 -943
-748 -651 9741 - 3 51 -1474 -1474
-1475 2185 13079 120556 778030 8722386 21154 3754 - 1 9 11 - 2 4 32
Tukey EXHIBIT
F)
2,
CONTINUED
SITUATION
16
BEST
1828 1797 19 39 1823 1829 1916 1918 2032 1986 1871 1874 1990 1944 2109 1919 1886 1865 19 32 2073 2231
1872 1930 2009 2016 2065 2045 2085 2070 2081 2085 2110 2120 2190 2192 2192 2249 2258 2229 2241 2335 Gt
H»
SITUATION
16
OAT A
20
STEP
2
FÜR
STEP
2
1841 1845 1959 1893 189C 1957 1946 2046 2018 1944 1965 2038 2015 2126 2023 1945 1952 2042 2137 2250
1821 1804 1937 1837 1841 1922 1912 2034 1983 1885 1892 1997 1956 2103 1940 1890 1883 1957 2085 2223
ROWS-PLUS-COLUMNS
ANALYSIS
2111
-5
-164
-161
-118
-231 -171 -95 -82 -86 -63 -41 -36 -22 -17 18 21 55 82 103 45 85 136 140 223
0 -2 0 -5 47 4 22 2 0 -1 -12 -5 30 5 -15 99 68 -11 -3 7
114 23 89 -39 -30 34 14 123 63 -56 -89 23 -56 81 -129 -104 -166 -149 -13 62
104 27 84 -28 -21 37 5 12 2 57 -45 -74 27 -47 72 -111 -103 -151 -127 -4 51
81 25 63 - 15 - 15 29 - 3 91 49 -29 -44 25 - 31 52 -71 -91 -125 - 85 4 35
SITUATION
16
STEP
3
1885 1948 2020 204C 2C18 2C49 2041 2078 2095 21 CO 2148 2143 2158 2194 2236 21 10 2144 22 64 22 68 2333
ROWS-T I M E S - C O L U M N S
ANALYSIS
OF
1 921 2C49 2C79 2189 2162 2145 2156 21 C7 2162 2269 2327 2245 23C5 2270 2449 2333 2380 2496 2393 2425 BEST
1951 2149 2134 2 3 36 23 34 2246 2299 2131 2225 2451 2499 2339 2471 2351 2652 2667 2685 2732 2513 2526
?029 ?i-I • CQ r*
i
1 i + +
+ + + + + + + + + + + + + 1 + + + + + + +
a o ÍH
• •M 1-1 Mi i— O u
(M m r^ co — iH i 1
uy
1 1 1 + +
+ 1 + + + + + + + + + + + 1 +
+ + 4+ + + + + + 1 + + + + + + + + + + + + + + + + + + + + O + +
CO vO r- i-H NÛ r- CTN ^ i vO in i
+
1 + + + + +
+ + + + + + + + + + + + 1 i + + + + + + + + + +
1
+ + + + + + + i + + +
+ + + + + + + + + + + + i i + + + + + + + + + + + + + + + + + +
1 i + +
1
it ON CM — CO o 00 vO 00 CO rONv£> o m i-H oo 00 i—i vO r^ CM CM i-H CM vO CM CM r- t^ ^ i i 1 1 1 i H CM
m CM vO m 00 00 CO ON r- CM m CO O vO CO i-H i-H m 00 xO vO O i-H CM CO m 00 ONvO i 1 1—1 i-H — i—i i1 i i i-H CM CM CM i i
i 1 CM i-H ^ 00 ON CM 00 ^ vO ^ (M m o m CM r- 00 CM i-H — rH i-H 00 i—i NO CO I-H CM CO 0 s in in O m in CM i-H CO i-H 1 1 i 1 i 1 1 1 i 1 i-H
t) PH ai ¡> o>
i ( CM CO vO CO vO i—i r^ CM i-H O CO o vO 1—1 CO CO i-H — 1—1 1 CO 1 1—1 1 i-H m m i—i i-H i-H 1 1 1 i i
u
1
d co S ^ h T3 3 £ •SP o h i
Off Response to Audio Tone
ACKNOWLEDGEMENTS T h i s study w a s supported in p a r t b y NIH G r a n t C o m p u t e r u s e in the H e a l t h S c i e n c e s , 5 P 0 7 P R 0 0 0 1 1 - 0 9 . M r . A. Angelone d e s i g n e d and built the s q u a r e wave tone s t i m u l u s g e n e r a t o r , the switching l o g i c so that the wave w a s cut off at the end of a full c y c l e and the i n t e r t r i a l i n t e r v a l t i m e c o n t r o l unit. He then loaned us h i s H i F i s t e r e o h e a d p h o n e s . G r e a t e r love of knowledge i s indeed r a r e ! M r . J . C. C h r i s t o p h e r w r o t e the p r o g r a m to digitize the data and the s o f t w a r e to g e n e r a t e the I B M 360 c o m p a t i b l e data t a p e . He a l s o d i g i t i z e d the d a t a .
349
Martin and Lucas DATA SET DESCRIPTION
DATA - EVOKED OFF RESPONSE TO AN AUDIO TONE (D. C. MARTIN AND H. L. LUCAS) CASES AND VARIABLES THERE ARE SEVEN SUBJECTS WITH THREE RUNS EACH. EACH RUN CONSISTS OF 75 TRIALS WITH 256 SAMPLES PER TRIAL. THE DATA ARE AVAILABLE ON A STANDARD LABELED 9—TRACK TAPE, BRIEF DESCRIPTION VAR
NAME
***
****
1 2 3 * 5 6 7 8 9 10 11 12-267
SUBJECT NUMBER AND RUN NUMBER NUMBER OF POINTS IN TRIAL STAT PACKAGE HEADER WORD STAT PACKAGE HEADER WORD STAT PACKAGE HEADER WORD PAD PAD TRIAL NUMBER NORMAL (TONE-ON) OR INVERTED (TONE-OFF) NUMBER OF INTERRUPTS SAMPLE RATE DATA POINTS
PURPOSE OF STUDY THE STUDY EXAMINED EVOKED ELECTRICAL POTENTIALS OF THE BRAIN TO AN AUDIO STIMULUS. ELECTRICAL POTENTIALS WERE AMPLIFIED BY A STANDARD GRASS MODEL 7 POLYGRAPH WITH 7T5 PREAMPLIFIERS. THE DATA WERE COLLECTED ON AN FM AMPEX ANALOGUE FR 1300 PORTABLE TAPE RECORDER AND A CONVENTIONAL INK STRIP CHART RECORDER. THREE CHANNELS OF DATA WERE COLLECTED. ONE CHANNEL WAS A MARKER CHANNEL TO INDICATE WHEN TO SAMPLE THE EEG. ONE CHANNEL WAS A TONE GENERATED BY THE STIMULUS GENERATOR. THE LAST WAS AN EEG CHANNEL. THE DATA RUNS ARE APPROXIMATELY SIX MINUTES LONG. THERE ARE SEVEN SUBJECTS WITH THREE RUNS EACH. EACH RUN CONSISTS OF 75 TRIALS WITH 256 SAMPLES PER TRIAL. REFERENCE MARTINt 0. C. AND H. L. LUCAS. FIRST TIME. DETAILED VAR
***
PUBLISHED
DESCRIPTION COL ***
FORMAT ******
DESCRIPTION ***********
350
IN THIS VOLUME FOR THE
Off Response to Audio Tone
2 3 4 5 6 7 8 9
10 11
2-10
xxxxxxxxx.
11- 19 2 0 - 28 2 9 - 37 38- 4 6 4 7 - 55 56- 64 6 5 - 73 7 4 - 82
XXXXXXXXX. XXXXXXXXX. XXXXXXXXX. XXXXXXXXX. XXXXXXXXX. XXXXXXXXX. XXXXXXXXX. XXXXXXXXX.
83-91 92-100
XXXXXXXXX. XXXXXXXXX.
SUBJECT NUMBER AND RUN NUMBER F I R S T D I G I T = SUBJECT NUMBER ( 1 - 7 ) SECONO D I G I T = RUN NUMBER ( 1 - 3 ) NUMBER OF POINTS IN T R I A L ( 2 5 6 ) STAT PACKAGE HEADER WORD ( 3 2 7 7 0 2 ) STAT PACKAGE HEADER WORD ( 5 0 ) STAT PACKAGE HEADER WORD ( 4 1 9 4 3 2 0 ) PAD ( 0 ) PAD ( 0 ) T R I A L NUMBER ( 1 - 7 5 ) NORMAL (TONE--ON) OR INVERTED ( T O N E - O F F ) 1 = NORMAL (TONE-ON) 2 = INVERTED ( T O N E - O F F ) NUMBER OF INTERRUPTS (1 M S ) / 6 SAMPLE RATE 1000 HZ
12
2-7
XXXXXX.
FIRST
30
110-115
XXXXXX.
N I N E T E E N T H DATA
DATA
POINT POINT
THE NEXT 13 RECORDS ARE IDENTICAL IN FORMAT TO THE PRECEDING ONE EXCEPT THAT THE LAST RECORD CONTAINS ONLY N I N E DATA P O I N T S . FORMAT
IS
LOCATION OF THE DATA DSL IB 1 .
(IX,UF9.0/13(1X,19F6.0/)
DATA ARE A V A I L A B L E
DATA **»* SUBJECT SUBJECT SUBJECT SUBJECT SUBJECT SUBJECT SUBJECT SUBJECT SUBJECT SUBJECT SUBJECT SUBJECT SUBJECT SUBJECT SUBJECT SUBJECT SUBJECT SUBJECT SUBJECT SUBJECT SUBJECT
tlX,9F6.0)
1. 1. It 2, 2* 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6» 6, 6, 7, 7, 7,
L I S T I N G OF DATA
ON A STANDARD L A B E L E D 9—TRACK
DSNAME ****** RUN RUN RUN RUN RUN RUN RUN RUN RUN RUN RUN RUN RUN RUN RUN RUN RUN RUN RUN RUN RUN
FOR
1 2 3 1 2 3 1 2 3 I 2 3 1 2 3 1 2 3 1 2 3
LABEL *****
SUB1RUN1 SUB1RUN2 SUB1RUN3 SUB2RUN1 SUB2RUN2 SUB2RUN3 S UB3RUN1 SUB3RUN2 SUB3RUN3 SUB4RUN1 SUB4RUN2 SUB4RUN3 SUB5RUN1 SUB5RUN2 SUB5RUN3 SUB6RUN1 SUB6RUN2 SUB6RUN3 SUB7RUN1 SUB7RUN2 SUB7RUN3
SUBJECT
l
f
TRIAL
351
14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
1
TAPE
LABELED
Martin and Lucas (THE
FIRST
FIVE
LINES
ARE
COLUMN
GUIDES)
S T U D I E S WITH A T R E M E N D O U S AMOUNT OF D A T A t S U C H AS T H I S O N E . MUST U T I L I Z E T A P E OR D I S K . THE S A M P L E OF DATA L I S T E D BELOW C O N S I S T S OF TWO T R I A L S FROM RUN 1 FOR S U B J E C T 1 . THE E N T I R E F I L E FOR T H I S S T U D Y I S A L M O S T 8 0 0 T I M E S AS L A R G E AS T H I S S A M P L E . 0 1 2 3 4 5 6 1234567890123456789012345678901234567890123456 78901234567890123456 1 1 7 8 9 0 1 7890123456789012345678901234567890123456789012345 256
11 -1425
-463
831
-837
-1633
-545
-3329
-3681
-3813
-561
-1415
-2089
-6159 -6947 -5633 -4407 -465 7 -4439 -2369 -2733
-2959
-2689 -1867
-469
511
343
1487
4991
4209
3315
1493
1881
2047
3199
3 543
2673
4095
4451
3427
2 869
1515
1913
327702 1 1971 1303 -257 319 -287 -1331 -2373 -1639 -4007 -3495 -2097 -2273 -2247 -2431 -3137 -4017 -5421 -5857 -5645 -5361 - 3 7 8 3 - 3 8 89 -2401 -2593 -2817 -2913 -1711 -599 -185 -1217 -701 -1137 2927 4591 4479 4847 3441 2921 1613 1407 2105 2047 3967 5231 3519 3761 1919 1727 3823 3389 1969 3627 2559 1663
50 2 1129 -571 -219 -1089 -2967 -2243 -2641 -2855 -6321 -4401 -3885 -2801 -2539 -199 -1619 -809 4847 5183 3001 2175 1407 3839 3199 2227 4479 4863 313
4194320 0 5089 1000 2679 2659 1489 -449 -335 -1285 -129 -2717 -3143 -1769 -2689 -3339 -2353 -2129 -1861 -2467 -2113 -1247 -2195 -2183 -2539 -2651 -2319 -1249 -4743 -4783 -4961 -5001 -5311 -5559 -3985 -3377 -3265 -2305 -2033 -2177 -2205 -2115 -2087 -971 -1153 -1373 -823 -385 -413 - 5 79 -719 -341 5311 4031 3953 4479 4415 4095 2623 2225 2337 2047 1623 1745 1249 1585 3071 3041 2047 1319 2047 1883 2407 3199 4095 3763 5567 5887 5103 4735 4991 5487 383 511 255
4194320 327702 50 0 2 5551 1000 2 3199 -979 2047 2639 4607 7599 6271 -577 1521 -913 -965 -151 369 1023 -481 1347 1727 1981 2175 745 1203 1991 1651 2297 2687 2641 3455 3967 3525 2473 2175 2175 1887 1407 1641 1383 1151 639 1241 -845 -1231 -1541 -1715 -1725 -691 -2735 -3131 -3613 -3521 -3985 -3765 -3985 -4865 -4953 -5123 -4509 -4331 -4043 -3873 -3905 -3377 -2999 -1927 -1667 -2889 -3045 -2775 -2707 -2457 -2545 -3161 -3521 -4145 -4503 -4395 -3483 -2753 -2997 -3137 -3313 -3527 -3249 -3201 -3249 -3343 -2017 -1409 -1689 -2619 -2757 747 3275 2337 -455 2639 22 73 3135 -1889 -1551 3429 2815 2793 2943 2303 2673 11
0 -163 -667 -2177 -3775 -1921 -1537 -1567 -2357 -5557 -4893 -3073 -2001 -1921 -1039 -1483 -257 4095 4463 2111 1779 2559 1131 1913 3 519 3481 4991
0
2 56
352
51 -77 -1817 -2993 -2389 -915 -2049 -4353 - 5 6 39 -5105 -2621 -1873 -2129 -1587 -1121 -283 4351 5439 2401 1469 2389 2429 1599 4095 1919 4479
1023 767 2291 2675 191 -2183 -4681 -4289 -2239 -4487 -3025 -1921 3017 2685
1279 975 2375 1787 -157 -2583 -4815 -4033 -2063 -4273 -3593 -2101 1983 2765
Off Response to Audio Tone 3839
3839
4575
3281
3581
2647
2785
2943
1879
1897
2343
1735
-385
-485
-641
3261
2929
1645
497
115
319
5359 4095 2417 26 73 1239 895 1149 -17 -305 895 431 1649 207
5183 3957 1851 2397 1535 1279 637 437 481 733 639 995 823
3125 3647 2559 2485 1373 1791 -129 369 1407 1535 383 255 735
353
2815 3665 3041 1625 1535 2453 63 -1 2289 1971 337 -705 613
3041 3709 3391 751 15 85 1407 -65 615 3433 3007 79 -371 959
3583 4351 2877 1279 1747 1225 -205 625 3327 2813 -1 73 1215
4479 4847 3071 1247 1791 1619 213 633 2267 1621 1717 767
4543 3929 2547 1633 1023 1279 187 -519 1279 2673 2257 1087
Martin and Lucas
REFERENCES B a r t l e t t , F . and E . R . John ( 1 9 7 0 ) . ments. Science 169, 3 0 4 - 3 0 5 .
R e p l y to S c h w a r t z ' s
com-
B i n g h a m , C . , M. D. G o d f r e y and J . W. T u k e y ( 1 9 6 7 ) . Modern t e c h n i q u e s of power s p e c t r u m e s t i m a t i o n . I. E . E . E . T r a n s a c t i o n s on Audio and E l e c t r o a c o u s t i c s , A U 1 5 , No. 2, 5 6 - 6 6 . Donchin, E . { 1 9 6 9 ) . D a t a a n a l y s i s t e c h n i q u e s in a v e r a g e evoked potential r e s e a r c h . In A v e r a g e E v o k e d P o t e n t i a l s , E . Donchin and D. B . L i n d s l e y , E d s . , W a s h i n g t o n , D. C . Govt. P r i n t i n g O f f i c e s , N A S A S P - 1 9 1 , 1 9 9 - 2 1 7 . F r e e m a n , W . J . (1964). prepyriform cortex.
A l i n e a r d i s t r i b u t e d f e e d b a c k m o d e l for E x p . Neurol. 10, 5 2 5 - 5 4 7 .
K i a n g , N . Y . S . and T . T . Sandel ( 1 9 6 1 ) . O f f - r e s p o n s e s f r o m the a u d i t o r y c o r t e x of u n a n e s t h e t i z e d c a t s . A r c h . Ital. B i o l . 99, 121-134. M o r s e , A. P . and F . E . Grubbs ( 1 9 4 7 ) . The e s t i m a t i o n of d i s p e r s i o n f r o m d i f f e r e n c e s . Ann. M a t h . S t a t i s t . 18, 1 9 4 - 2 1 4 . Sandel, T . T . and N. Y . S. Kiang ( 1 9 6 1 ) . a u d i t o r y c o r t e x of a n e s t h e t i z e d c a t s . 105-120.
O f f - r e s p o n s e s f r o m the Arch. Ital. Biol. 99,
S c h w a r t z , M . ( 1 9 7 0 ) . M e a n s and v a r i a n c e s of a v e r a g e - r e s p o n s e wave f o r m s . S c i e n c e 1 6 9 , 3 0 3 - 3 0 4 .
354
DISCUSSION O F T H E L U C A S AND M A R T I N C H A P T E R
R.
Mickey
P r o f e s s o r s M a r t i n and L u c a s have p r e s e n t e d us with a l a r g e d a t a s e t f r o m a c a r e f u l l y planned e x p e r i m e n t . It even a p p e a r s that t h e y applied to t h e m s e l v e s the advice t h e y would give to o t h e r s , p a r t i c u l a r l y with r e s p e c t to advance planning of the a n a l y s i s . U n f o r t u n a t e l y — in one s e n s e at l e a s t — t i m e did not p e r m i t the plans to c o m e to f r u i t i o n . On the o t h e r hand, as i l l u s t r a t i v e d a t a f o r a n a l y s i s this i n c o m p l e t e n e s s l e a v e s us with f r e e s c o p e to s p e c u l a t e on what m i g h t be i n t e r e s t i n g to t r y , without being i n h i b i t e d b y knowing what w o r k e d out w e l l . I a g r e e with M a r t i n and L u c a s that the p r o b l e m i s r e p r e s e n t a t i v e of a s u b s t a n t i a l c l a s s ; it p e r h a p s h a s the advantage of being m o r e " n e a t " than m a n y s i m i l a r p r o b l e m s in that it m a y not p r e s e n t as m a n y side i s s u e s to d i v e r t our a t t e n t i o n . One q u e s t i o n that a r i s e s f o r m e i s w h e t h e r or not t h e r e i s in f a c t a r e s p o n s e . I f e e l a bit u n c o m f o r t a b l e in a s k i n g the q u e s t i o n , s i n c e t h o s e in the evoked r e s p o n s e b u s i n e s s d o u b t l e s s have m e t h o d s of d e t e r m i n i n g t h i s that a r e a c c e p t a b l e to the c o m m u n i t y . Or i s t h i s p o s s i b i l i t y one of the q u e s t i o n s f o r us to t r e a t in the d a t a a n a l y s i s ? T h e q u e s t i o n of r e s p o n s e m a y not be s e n s i b l e to a s k of a given t r i a l o r r u n , although i t would s e e m n a t u r a l to s a y of a given run that the s u b j e c t showed a r e s p o n s e , p a r t i c u l a r l y if the r e s p o n s e was l a r g e in s o m e s e n s e . One of the r e a s o n s f o r r a i s i n g the q u e s t i o n at all c o m e s f r o m wondering if one d o e s n ' t n e e d p r e - s t i m u l u s data to give an a n s w e r . P e r h a p s the q u e s t i o n i s a side i s s u e . What is a v a i l a b l e to look at in the way of r e s u l t s i s the s e t of t r a c i n g s of p o s t - s t i m u l u s a v e r a g e p o t e n t i a l for e a c h of the t h r e e r u n s for e a c h of the s e v e n t e s t s u b j e c t s . B y way of s u b j e c t i v e a n a l y s i s , I r e m o v e d the t r e a t m e n t and run n u m b e r i d e n t i f i c a t i o n and p r e s e n t e d the t r a c i n g s to a couple of s u b j e c t s , t e l l i n g t h e m that f o r e a c h t e s t two of the t r a c i n g s followed one s t i m u l u s and the t h i r d t r a c i n g followed a s e c o n d s t i m u l u s ; the s u b j e c t s w e r e 355
Martin and Lucas, discussion
a s k e d to pick the one t h a t w a s d i f f e r e n t . The t a s k a p p e a r e d to p r e s e n t no d i f f i c u l t i e s and b o t h s u b j e c t s c o r r e c t l y s o r t e d the t r a c i n g s f o r e a c h of t h e s e v e n t e s t s . T h e r e p l i c a t e t r a c i n g s s e e m to be quite s i m i l a r and a r e s u f f i c i e n t l y d i f f e r e n t f r o m t h e t h i r d that t h e r e w a s no point in continuing with additional s u b j e c t s . The r e p l i c a t e t r a c i n g s a r e a l w a y s the s e c o n d and t h i r d r u n s , so it is p o s s i b l e t h a t the d i s t i n c t i o n is an o r d e r r a t h e r t h a n a t r e a t m e n t e f f e c t . The s u b j e c t s w e r e then a s k e d to s o r t the t r a c i n g s into two g r o u p s to c o r r e s p o n d to the two t r e a t m e n t s . One s u b j e c t s o r t e d the t r a c i n g s in c o m p l e t e c o r r e s p o n d e n c e with t r e a t m e n t and t h e o t h e r h a d a f e w e r r o r s . The s e c o n d t a s k h a s s u b s t a n t i a l l y m o r e u n c e r t a i n t y t h a n t h e f i r s t b e c a u s e of t h e v a r i a b i l i t y among t e s t s u b j e c t s , but n e v e r t h e l e s s does not s e e m to p r e s e n t a g u e s s i n g s i t u a t i o n . M y c o n c l u s i o n i s t h a t a t l e a s t o n e of t h e s t i m u l i e l i c i t s a r e s p o n s e a n d , a l t h o u g h t h e r e i s q u i t e a b i t of v a r i a t i o n a m o n g r e s p o n d e r s , t h e r e s p o n s e p a t t e r n s a m o n g t e s t s u b j e c t s a r e s i m i l a r enough to g e n e r a l l y d i s t i n g u i s h the two stimuli. I also t r i e d o r d e r i n g the t e s t s u b j e c t s by r e s p o n s e p a t t e r n s . This w a s d o n e s e p a r a t e l y f o r t h e o n r e s p o n s e a n d t h e off r e s p o n s e . My object was to group together p a t t e r n s that s e e m e d quite s i m i l a r a n d t o s e e if t h e r e w a s a g r a d a t i o n a m o n g t h e p a t t e r n s . I d i d t h i s s e p a r a t e l y f o r t h e o n r e s p o n s e a n d t h e off r e s p o n s e . T h e r e w a s n o r a n k c o r r e l a t i o n b e t w e e n t h e t w o o r d e r i n g s of t h e s u b j e c t s . While t h e r e w e r e s o m e t r a c i n g s that s e e m e d r e l a t i v e l y quite s i m i l a r , (for e x a m p l e , the on r e s p o n s e for s u b j e c t s 1 a n d 4) i n g e n e r a l I d o u b t t h a t m y o r d e r i n g s w o u l d b e v e r y r e p r o d u c i b l e w e r e I to r e p e a t the p r o c e s s after a suitable t i m e lapse. T h e r e s u l t s of o n e s u c h o r d e r i n g a r e s h o w n i n f i g u r e 1. I d o n ' t t h i n k t h e r e i s any p a r t i c u l a r s i g n i f i c a n c e to the o r d e r i n g , o r e v e n t h a t a t t e m p t i n g t o o r d e r i s p a r t i c u l a r l y i n s t r u c t i v e . I do t h i n k t h e f i g u r e b r i n g s o u t s i m i l a r i t i e s a m o n g p a t t e r n s and m a y be u s e f u l for thinking about n u m e r i c a l analysis. F o r e x a m p l e , t h e r e a p p e a r s to be an i m p o r t a n t a m p l i t u d e v a r i a t i o n a m o n g s u b j e c t s . A l s o t h e r e is s o m e s u g g e s t i o n t h a t s u b j e c t s v a r y in
356
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OFF
Subject
4
RESPONSE x Run
ON
3
RESPONSE
Subject
/
Run I
Subject
4
Run l
Run 2
Subject 2
Run 1
Subject
Run 1
6
Subject 2
,Run 3 Run 2
Subject
/
Subject 6
Run 2
Run 2
Run 3
Subject 3
Run 1
Subject 5
Run 1
F i g u r e 1. E v o k e d off and on r e s p o n s e s to a n a u d i o tone a r r a n g e d a c c o r d i n g to g r a d a t i o n of r e s p o n s e . T r a c i n g s a r e f r o m t h o s e s u p p l i e d by L u c a s and M a r t i n . T h e r e is no d i f f i c u l t y in s o r t i n g out the r e s p o n s e s of a g i v e n s u b j e c t to the r e p e a t e d s t i m u l u s , and l i t t l e to m o d e r a t e d i f f i c u l t y in d i s t i n g u i s h i n g b e t w e e n on and off r e s p o n s e in an u n l a b e l e d p r e s e n t a t i o n of the response curves.
357
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t h e i r time scale. R e p r e s e n t a t i o n s that explicitly allow for these v a r i a t i o n s m i g h t be m o r e u s e f u l t h a n r e p r e s e n t a t i o n s t h a t do n o t . I t a k e t h e r e s u l t s a s shown in f i g u r e 1 to i n d i c a t e t h a t t h e r e p r o b a b l y a r e s o m e n i c e l y i n f o r m a t i v e w a y s of a n a l y z i n g t h e d a t a , and I l o o k f o r w a r d to the c o m p l e t e d a n a l y s i s b y M a r t i n and L u c a s . O n e of t h e d i f f i c u l t i e s w i t h t h i s t y p e of p r o b l e m s e e m s t o b e t h a t d e s p i t e a l l of t h e n u m b e r s t h a t e n t e r t h e d a t a b a s e , t h e r e a r e n o t m a n y d e g r e e s of f r e e d o m w h e n i t g e t s d o w n t o d e v e l o p i n g t h e f i n a l c o n c l u s i o n s . T h i s a s p e c t of t h e a n a l y s i s s e e m s m o r e s e v e r e if one w e r e to take into account other a t t r i b u t e s (such as h e a r i n g s e n s i t i v i t y , p i t c h d i s c r i m i n a t i o n a b i l i t y , and p e r h a p s i n d i c a t o r s of n e r v o u s s e n s i t i v i t y ) t h a t m i g h t p l a u s i b l y r e l a t e t o t h e r e s p o n s e . S i n c e c o v a r i a t e t y p e i n f o r m a t i o n i s n o t a v a i l a b l e w e do n o t n e e d t o c o n s i d e r h o w it m i g h t a f f e c t the a n a l y s i s . Noting t h e p o s s i b i l i t y d o e s r a i s e t h e q u e s t i o n of t h e e x t e n t t o w h i c h t h e d a t a a n a l y s i s n e e d s t o b e b a s e d o n i d e a s of p h y s i o l o g i c a l i n t e r e s t . I w i l l h a z a r d t h e o p i n i o n t h a t d a t a a n a l y s t s c a n d e v e l o p f i n d i n g s of p h y s i o l o g i c a l i n t e r e s t w i t h a m i n i m u m of g u i d a n c e f r o m p h y s i o l o g y a n d t h a t i t i s q u i t e a p p r o p r i a t e t o p r o c e e d o n o u r o w n . In o r d e r to c o n t r i b u t e e f f e c t i v e l y to t h e s c i e n t i f i c s u b s t a n c e , h o w e v e r , I think that a s e c o n d , t h i r d , e t c . p a s s at the d a t a w i l l o r d i n a r i l y be n e e d e d , and that t h e s e will r e q u i r e a p p r o p r i a t e c o l l a b o r a t i o n .
Jukey (preconference
comments)
The m a i n i s s u e s h e r e s e e m to be the e x t e n t to which b a c k g r o u n d i n t e r f e r e n c e (mainly f r o m alpha r h y t h m s ) can be allowed f o r , and w h a t o u g h t to b e d o n e t o l o o k m o r e c l e a r l y i n t o t h e d a t a , e s p e c i a l l y i n v i e w of t h e p o s s i b i l i t y of e n t r a i n m e n t of a l p h a r h y t h m s . L e t u s s u p p o s e t h a t w e h a v e b e e n a b l e t o d i g i t i z e a l o n g p i e c e of r e c o r d for m a n y c a s e s . F o r definiteness I suppose details that m a y not be e s s e n t i a l , s p e c i f i c a l l y that we have d a t a for a d j a c e n t t i m e intervals as follows:
358
Off Response to Audio Tone, discussion
I n t e r v a l A = 256 p o i n t s at 8 m i l l i s e c o n d s I n t e r v a l B = 256 p o i n t s at 1 m i l l i s e c o n d (stimulus change) I n t e r v a l D = 256 p o i n t s a t 1 m i l l i s e c o n d T h e f i r s t s t e p i s to a n a l y z e i n t e r v a l s A a n d B t o g e t h e r , s e e k i n g f o r a good p r e d i c t o r a t e a c h p o i n t of i n t e r v a l B b a s e d o n t h e v a l u e s i n i n t e r v a l A. A s a p r a c t i c a l m a t t e r i t m a y w e l l s u f f i c e t o f i n d r e g r e s s i o n s f o r p o i n t s of i n t e r v a l B s p a c e d e v e r y 8 t o 16 m i l l i s e c o n d s and i n t e r p o l a t e b e t w e e n . F i n d i n g a s m o o t h e d m e a n s p e c t r u m f o r i n t e r v a l A will guide us to f a i r l y good l i n e a r p r e d i c t o r s , a n d s t e p w i s e a d j u s t m e n t ( u s i n g b o t h r a w and l o c a l l y smoothed values as possible c a r r i e r s ) can polish these p r e d i c t o r s up easily. L e t n o w i n t e r v a l C c o n s i s t of 256 p o i n t s at 8 m i l l i s e c o n d s p a c i n g r u n n i n g u p t o j u s t b e f o r e t h e s t i m u l u s c h a n g e . It w i l l c o n t a i n t h e l a s t 256 - 32 = 2 2 4 p o i n t s of i n t e r v a l A e v e r y 8 t h p o i n t , 32 i n a l l , of i n t e r v a l B . We c a n n o w a p p l y t h e s a m e p r e d i c t o r s t o i n t e r v a l C t h a t w e f o u n d d i d w e l l w h e n a p p l i e d t o i n t e r v a l A. T h e y m a y o r m a y n o t do w e l l i n s u p p r e s s i n g b a c k g r o u n d o v e r i n t e r v a l D. (If w e h a d n o t changed the s t i m u l u s , they would have done well. ) Since we a r e in doubt a s to how t h e s e p r e d i c t o r s w i l l w o r k , it is n a t u r a l t o t r e a t t h e m a s c o v a r i a t e s , and a s k w i t h w h a t c o e f f i c i e n t s t h e y w i l l do t h e i r b e s t . T h e s e c o e f f i c i e n t s m i g h t w e l l c h a n g e a s w e m o v e a c r o s s i n t e r v a l D. If t h e p a t t e r n of t h e i r change is c l e a r enough, t h i s p a t t e r n ought to t e l l us m o r e about w h a t is r e a l l y going on. Plausibility a r g u m e n t s can be m a d e f o r trying both " p r e d i c t o r v a l u e " a n d " s m o o t h e d r a t e of c h a n g e of p r e d i c t o r v a l u e w i t h t i m e p r e d i c t e d f o r " a s c o v a r i a t e s . (An e m p i r i c i s t m i g h t a l s o b e w i l l i n g to d u m p i n t h e s q u a r e of t h e l a t t e r . ) 359
Martin and Lucas, discussion
In a n y e v e n t , c o v a r i a n c e , a c r o s s t r i a l s b u t w i t h i n i n d i v i d u a l s , o u g h t t o b e t r i e d a s a w a y to g e t b e t t e r d a t a f o r t h e f i n a l a n a l y s i s . Another, m o r e complex h y p o t h e s i s , would be that the stimulus c h a n g e e n t r a i n s t h e a l p h a r h y t h m s . A w a y t o t r y to u s e t h i s would be the following: •
m a k e predictions through i n t e r v a l D as b e f o r e ;
•
find t h e i r F o u r i e r t r a n s f o r m s ;
•
r o t a t e e a c h ( c o s i n e c o e f f , s i n e c o e f f ) 2 - v e c t o r i n two w a y s , so t h e r e s u l t i n g c o v a r i a t e (a) h a s a + m a x i m u m at t h e s t i m u l u s c h a n g e , (b) h a s a + m a x i m u m s l o p e there;
•
t r a n s f o r m e a c h of (a) and (b) b a c k ; and
•
u s e t h r e e c o v a r i a t e s , t h e o r i g i n a l p r e d i c t o r and t h e r e t r a n s f o r m s of (a) and (b).
If t h i s d o e s b e t t e r , i t i s t h e r e s i d u a l s f r o m t h i s t h a t o u g h t t o go to f u r t h e r analysis. What f u r t h e r a n a l y s i s ? It s e e m s t o m e t h a t a D F T ( d i s c r e t e F o u r i e r t r a n s f o r m ) s h o u l d c o m e f i r s t . We k n o w t h a t d i f f e r e n t f r e q u e n c y b a n d s b e h a v e q u i t e d i f f e r e n t l y . We o u g h t t o h a v e a l o o k at w h a t w e c a n do a b o u t t h i s f i r s t . W h i l e i t w o u l d b e p o s s i b l e t o D F T t h e 256 v a l u e s (which m a y h a v e b e e n a d j u s t e d by c o v a r i a n c e ) d i r e c t l y , which c o r r e s p o n d s to u s i n g a r e c t a n g u l a r d a t a w i n d o w , and w h i l e it m a y be w o r t h w h i l e t o do t h i s f o r c o m p a r i s o n , p a s t e x p e r i e n c e s u g g e s t s s t r o n g l y t h a t t h i s w i l l n o t t u r n o u t t o b e t h e b e t t e r t h i n g to d o . D F T u s i n g a d a t a w i n d o w of t h e f o r m flat at 1 p i e c e s of c o s i n e data here 360
Off Response to Audio Tone, discussion
with the ends about 2 0 m i l l i s e c o n d s long is m u c h m o r e likely to give c l e a n r e s u l t s . What do we do with o u r sine and c o s i n e c o e f f i c i e n t s ? Having obtained t h e m f o r m a n y r u n s f o r a single individual, we c a n think of plotting the s i n e - a n d - c o s i n e - c o e f f i c i e n t v e c t o r s f o r e a c h f r e q u e n c y . We will get s o m e kind of point cloud whose c h a r a c t e r i s t i c s will change f r o m f r e q u e n c y to f r e q u e n c y . One s i m p l e a p p r o a c h t h a t could be t a k e n would be to a s k what shrinking of the m e a n v e c t o r of e a c h point cloud l e a d s to m i n i m u m m e a n s q u a r e e r r o r . A f t e r finding the shrunken v a l u e s , t h e y c a n be r e t r a n s f o r m e d to a t i m e function that m i g h t be a b e t t e r t y p i c a l p a t t e r n f o r an individual than the a c t u a l t i m e m e a n s ( t h e m s e l v e s a r e t r a n s f o r m a t i o n of the u n s h r u n k e n m e a n v e c t o r s ) . Another a p p r o a c h would be to r e l a t e the D F T c o m p o n e n t s f o r i n t e r v a l D with t h o s e f o r i n t e r v a l B . H e r e r e l a t i o n s h i p could mean: •
o r d i n a r y ( v e c t o r ) r e g r e s s i o n , which would take c a r e of u n u s u a l d e p e n d e n c e s of p h a s e shift on f r e q u e n c y ;
•
r e g r e s s i o n on (or a l s o on) the amplitude of the e a r l i e r v e c t o r ; and
•
s o r t i n g out, s a y into 3 o r 4 e q u i n u m e r o u s p a r t s , in a c c o r d with the v a l u e s of the e a r l i e r a m p l i t u d e , followed by looking at s p r e a d s to s e e if weighting by s o m e function of the e a r l i e r amplitude should be helpful.
W h e t h e r o r not s o m e t i m e - s i d e function e x p a n s i o n i s to be u s e d , it s e e m s l i k e l y t h a t an i n i t i a l e x c u r s i o n to the f r e q u e n c y side will be w e l l w o r t h w h i l e . A n o t h e r p o s s i b i l i t y f o r the p r e d i c t i o n g a m e , which would be m o r e c o m p a c t , and m i g h t be m o r e e f f e c t i v e , would be to p r e d i c t s h o r t p e r i o d ( i n t e r v a l B o r D) F o u r i e r c o e f f i c i e n t s f r o m long p e r i o d
361
Martin and Lucas, discussion
( i n t e r v a l A o r C) F o u r i e r c o e f f i c i e n t s . H e r e only about 8 o r 10 (or m a y b e 16 o r 20) 2 - v e c t o r s would n a t u r a l l y a p p e a r a s c a r r i e r s f o r a g i v e n 2 - v e c t o r , and the s a m p l e s i z e c o u l d o f t e n be d o u b l e d b y 90° r o t a t i o n s . D i f f e r e n c e s in r e g r e s s i o n c o e f f i c i e n t s f o r B on A and D on C c o u l d be a s s i g n e d to " e n t r a i n m e n t " and m i g h t t h r o w a d d i t i o n a l l i g h t on the p r o b l e m . ( l a t e r c o m m e n t s , a f t e r r e a d i n g f i n a l v e r s i o n of p a p e r ) 1. We h a v e not yet s e e n e i t h e r m i d d l e o r l a t e r s t a g e s of a n a l y s i s for this p r o b l e m . 2. I w o u l d l i k e , w e r e I a n a l y z i n g the d a t a , to s e e p l o t s of " v a r i a n c e about m e a n " a g a i n s t " t i m e a f t e r s t i m u l u s " and a g a i n s t " s l o p e of m e a n . " (And the l a t t e r f o r a l l but the f i r s t p a r t of the r e s p o n s e . ) H a r d l o o k s at t h e s e m i g h t l e a d u s to s o m e i n s i g h t s a s to p l a u s i b l e m o d e l s (e. g . , a t i m e s t r e t c h - s h r i n k c o m p o n e n t in the r e s p o n s e ought to show u p a s i n c r e a s e d v a r i a n c e n e a r h i g h slopes). 3. A l l of u s — u n w i s e s t a t i s t i c i a n s t h a t we a r e — would l i k e to l o o k at the f u l l p r i n c i p a l c o m p o n e n t a n a l y s i s of d i m e n s i o n 2 5 6 . ( E x c e p t that one w o u l d n e e d about 400 r u n s , one c o u l d think of d o i n g t h i s . I h a v e done e i g e n v a l u e s in 104 d i m e n s i o n s . ) Why not p l a n to do s o m e t h i n g l e s s than t h i s , but m o r e f l e x i b l e than f i t t i n g c h o s e n s e t s of f u n c t i o n s ? One a p p r o a c h would be to t a k e a s t e p w i s e a p p r o a c h . Consider t i m e s l 6 h + 1, l 6 h + 2 , . . . , l 6 h + 16 f o r h = 0, 1, 2 , . . . , 15. F o r e a c h h s e p a r a t e l y , l o o k at the d e v i a t i o n s of the 75 r u n s f r o m t h e i r m e a n (or the t h r e e s e t s of 2 5 , e a c h f r o m i t s own m e a n s , e t c . ) and do a f u l l p r i n c i p a l c o m p o n e n t s a n a l y s i s . H o p e f u l l y 0 to 3 c o m p o n e n t s w i l l be o u t s t a n d i n g at e a c h h. If 0, we w i l l i n c l u d e the l a r g e s t a v a i l a b l e , g i v i n g u s 1 to 3. T h e s e c o n d s t e p i s to c o m b i n e the 16 to 40 r e s u l t i n g c o m p o n e n t s in a n o t h e r p r i n c i p a l c o m p o n e n t a n a l y s i s . The t r u l y o u t s t a n d i n g c o m p o n e n t s h e r e , if a n y , c a n b e t a k e n a s r e a s o n a b l e l e a d s .
362
Off Response to Audio Tone, discussion
G i v e n one such, we c a n s e p a r a t e the t r i a l s into 3 to 5 z o n e s b a s e d o n t h e v a l u e of t h i s c o m p o n e n t . T h e m e a n s of t h e z o n e s — m o r e p r e c i s e l y the d i f f e r e n c e s of, or a p p r o p r i a t e c o n t r a s t s among, t h e s e m e a n s — c a n g i v e u s t h e n a n e v e n b e t t e r p i c t u r e of w h i c h c o m p o n e n t s s e e m to show l a r g e v a r i a n c e (within s u b j e c t ) . An a l t e r n a t i v e t o 16 b l o c k s of 16 l a i d e n d t o e n d i s 15 b l o c k s of 32 l a i d " 1 6 t o w e a t h e r " (in s h i n g l e t e r m i n o l o g y ) , t h a t i s I 6 h + 1 , l 6 h + 2, . . . , I 6 h + 32 f o r h = 0, 1, . . . , 1 5 . In e i t h e r c a s e , t h e s t e p w i s e c h a r a c t e r of t h e p r i n c i p a l c o m p o n e n t s w o u l d k e e p u s f r o m looking at two w e i r d c o m p o n e n t s . (The l i m i t e d r e s u l t s of t h e s t e p w i s e e i g e n i n g m a y w e l l p r o v e s a f e r t h a n a f u l l 2 5 6 - d i m e n s i o n a l a n a l y s i s f o r f e w e r t h a n , s a y , 2000 t r i a l s . Who k n o w s ? ) 4 . T h e t r a c i n g of 6 0 H z s y n c h r o n i s m t o t h e s t i m u l u s c o n t r o l offers a simple design opportunity for later experiments. A r e v e r s i n g s w i t c h in the p o w e r l e a d f o r t h e c o n t r o l only should a l l o w l o c k i n g - i n t h e 60 H z i n r e v e r s e d p h a s e o n h a l f t h e t r i a l s , so that its e f f e c t will c a n c e l out in the m e a n . 5. D r . J o h n H a r t i g a n ' s c o m m e n t s f r o m "the p a t t e r n r e c o g n i t i o n p o i n t of v i e w " t a c i t l y a s s u m e t h a t t h e f e a t u r e s of i m p o r t a n c e w i l l show m o s t c l e a r l y in a t i m e h i s t o r y . I see no r e a s o n to a s s u m e t h i s — a n d , I g u e s s , n o r e a s o n t o a s s u m e t h e c o n t r a r y . It i s a n e v e r - p r e s e n t d a n g e r of d a t a a n a l y s i s t o a s s u m e t h a t , if t h e d a t a would like to show u s s o m e t h i n g , we can see it in the f o r m f i r s t p r e s e n t e d t o u s . ( T h e g r e a t u s e f u l n e s s of p o l y n o m i a l f i t s , a n d r o w - P L U S - c o l u m n a n a l y s e s of 2 - w a y t a b l e s of r e s p o n s e s a r e p r o m i n e n t d e n i a l s of a n y s u c h a s s u m p t i o n . )
363
CHAPTER 8 T H E USE O F D E N S I T Y E S T I M A T E S B A S E D ON O R T H O G O N A L EXPANSIONS
R I C H A R D A.
KRONMAL
D e p a r t m e n t of B i o s t a t i s t i e s , U n i v e r s i t y of W a s h i n g t o n ,
Seattle
and MICHAEL TARTER D i v i s i o n of B i o s t a t i s t i c s ,
U n i v e r s i t y of C a l i f o r n i a ,
Berkeley
In t h i s c h a p t e r w e d i s c u s s t h e a p p l i c a t i o n of o r t h o g o n a l p o l y n o m i a l e s t i m a t e s of t h e u n i v a r i a t e a n d b i v a r i a t e d e n s i t i e s a s w e l l a s t h e c u m u l a t i v e f o r a s e t of d a t a c o n t r a s t i n g c o r o n a r y h e a r t d i s e a s e p a t i e n t s w i t h a g e m a t c h e d " n o r m a l s . " We a r e c o n c e r n e d p r i m a r i l y w i t h t h e u s e of t h e s e t e c h n i q u e s f o r d e s c r i p t i o n of t h e d a t a , p a r t i c u l a r l y t h e i r u s e as p o s s i b l e r e p l a c e m e n t s f o r the h i s t o g r a m , two v a r i a b l e plot and s t e p function. A d e s c r i p t i o n of t h e s t u d y i s g i v e n b e l o w , a n d a d a t a s e t d e s c r i p t i o n t a k e n f r o m a p u b l i s h e d p a p e r d e s c r i b i n g t h e r e s u l t s of t h e s t u d y , i s g i v e n a t t h e e n d of t h e c h a p t e r . T h e r e s u l t s of t h e u s e of t h e s e m e t h o d s o n t h e d a t a a r e d e s c r i b e d , w i t h p a r t i c u l a r a t t e n t i o n g i v e n t o t h e e s t i m a t i o n of t h e c u m u l a t i v e a n d of t h e b i v a r i a t e d i s t r i b u t i o n s of s e v e r a l of t h e v a r i a b l e s . The general theory for orthogonal polynomial density estimation f o r t h e o r t h o n o r m a l s y s t e m b a s e d on t h e c o m p l e x t r i g o n o m e t r i c functions is also given.
365
Kronmal and Tarter
DESCRIPTION O F T H E STUDY This analysis is b a s e d on a study conducted by K a s s e r and B r u c e (1969) c o m p a r i n g a group of c o r o n a r y h e a r t d i s e a s e patients with a group of age m a t c h e d n o r m a l s . The d e s c r i p t i o n given below is taken f r o m the published paper with s e c t i o n s omitted for b r e v i t y . C o r o n a r y h e a r t d i s e a s e exhibits m a r k e d v a r i a b i l i t y in its c l i n i c a l m a n i f e s t a t i o n s . While s o m e patients a r e disabled by angina p e c t o r i s , o t h e r s surviving m y o c a r d i a l i n f a r c t i o n pursue t h e i r usual a c t i v i t i e s with little or no s y m p t o m s . N e v e r t h e l e s s , h e m o d y n a m i c studies indicate that m o s t c o r o n a r y patients have evidence of i m p a i r e d m y o c a r d i a l c o n t r a c t i l i t y . When s t r e s s e d by e x e r c i s e or i s o p r o t e r e n o l i n fusion, they show s u b n o r m a l i n c r e m e n t s in c a r d i a c output, stroke volume, and s y s t o l i c e j e c t i o n r a t e . M o r e advanced d i s e a s e m a y m a n i f e s t o v e r t left v e n t r i c u l a r f a i l u r e with elevated r e s t i n g left v e n t r i c u l a r e n d - d i a s t o l i c and p u l m o n a r y capillary pressures. In a p a r a l l e l m a n n e r , advancing age a l s o a d v e r s e l y a f f e c t s c a r d i o v a s c u l a r function. W h e r e a s older individuals show only slight a b n o r m a l i t i e s at s u b m a x i m a l work l o a d s , t h e i r m a x i m a l e x e r c i s e p e r f o r m a n c e is significantly l i m i t e d . Oxygen consumption, h e a r t r a t e , s t r o k e v o l u m e , and c a r d i a c output a r e all r e d u c e d at high w o r k l o a d s . Consequently, m o s t m i d d l e - a g e d c o r o n a r y patients suffer f r o m the added limitations of both d i s e a s e and i n c r e a s i n g age. Many c l i n i c i a n s l i m i t e x e r c i s e testing of c o r o n a r y patients to simply detecting m y o c a r d i a l i s c h e m i a f r o m p o s t e x e r t i o n a l S - T d e p r e s s i o n . Y e t with slightly m o r e e f f o r t , an objective a s s e s s m e n t of e a c h patient's functional c a p a c i t y can be obtained. In this r e g a r d , m a x i m a l e x e r c i s e on a m u l t i s t a g e t r e a d m i l l has proved to be a useful t e s t . In l e s s than 12 m i n . , a s y m p t o m a t i c as well as a c l a s s IV c a r d i a c patients (New Y o r k H e a r t A s s o c i a t i o n c l a s s i f i c a t i o n ) can be e x e r c i s e d usirtg the s a m e t e s t p r o c e d u r e , and t h e i r p e r f o r m a n c e c o m p a r e d quantitatively to that of n o r m a l s u b j e c t s .
366
Density Estimates
This study w a s u n d e r t a k e n to define the m a x i m a l e x e r c i s e c a p a c i t y of n o r m a l m e n a n d c o r o n a r y p a t i e n t s a n d a s s e s s t h e r e l a t i v e c o n t r i b u t i o n s of a g i n g a n d d i s e a s e t o t h e o v e r a l l f u n c t i o n a l l i m i t a t i o n s of a n i n d i v i d u a l p a t i e n t . . . . O n e h u n d r e d s e v e n t e e n (117) m e n , w h o s e a g e s r a n g e d f r o m 34 t o 7 3 y e a r s w i t h a m e a n a g e of 52 y e a r s w e r e s e l e c t e d b e c a u s e of a c l i n i c a l d i a g n o s i s of d e f i n i t e c o r o n a r y h e a r t d i s e a s e . T h e d i a g n o s i s w a s b a s e d o n (1) r e c u r r e n t e x e r t i o n a l c h e s t p a i n t y p i c a l of a n g i n a p e c t o r i s , (2) a Q w a v e of at l e a s t 0. 0 4 - s e c d u r a t i o n on a r e s t i n g e l e c t r o c a r d i o g r a m , o r (3) a c l i n i c a l e p i s o d e c o m p a t i b l e w i t h a c u t e m y o c a r d i a l infarction substantiated by elevated serum enzymes or myoc a r d i a l i n j u r y r e c o r d e d on t h e e l e c t r o c a r d i o g r a m , o r b o t h . All m e n w e r e in s i n u s r h y t h m , and none w e r e r e c e i v i n g d i g i t a l i s . No p a t i e n t w a s e x e r c i s e d e a r l i e r t h a n 2 m o n t h s following a m y o c a r d i a l infarction. F o r c o m p a r i s o n , 117 a g e - m a t c h e d n o r m a l m e n , w h o s e a g e s r a n g e d f r o m 33 t o 7 2 , w i t h a m e a n a g e of 51 y e a r s , w e r e c h o s e n f r o m a g r o u p of h e a l t h y f a c u l t y a n d Y M C A v o l u n t e e r s . In a d d i t i o n , a t h i r d g r o u p of 6 2 n o r m a l y o u n g m e n , w h o s e a g e s r a n g e d f r o m 15 t o 35 y e a r s , w i t h a m e a n of 25 y e a r s , w e r e e v a l u a t e d in the o u t p a t i e n t c l i n i c . B o t h t h e s e g r o u p s of m e n w e r e j u d g e d t o b e n o r m a l b y h i s t o r y , p h y s i c a l e x a m i n a t i o n , c h e s t x - r a y s , and 1 2 - l e a d e l e c t r o c a r d i o g r a m . All n o r m a l s u b j e c t s w e r e n o r m o t e n s i v e a c c o r d i n g to World H e a l t h O r g a n i z a t i o n c r i t e r i a , with a r e s t i n g blood p r e s s u r e of l e s s t h a n 1 6 0 / 9 5 m m H g . D a t a w e r e a l s o a v a i l a b l e on 79 of t h e o l d e r n o r m a l m e n w h o h a d b e e n t e s t e d p r e v i o u s l y . The m u l t i s t a g e e x e r c i s e t e s t involves an u n i n t e r r u p t e d s e r i e s of w o r k l o a d s on a m o t o r d r i v e n t r e a d m i l l ; t h e i n i t i a l s u b m a x i m a l l o a d ( s t a g e 1) r e q u i r e s w a l k i n g s l o w l y o n a 10% g r a d e . S i n c e t h e r e i s no i n c r e a s e in o x y g e n u p t a k e a f t e r 3 m i n . of s u b m a x i m a l e x e r t i o n , t h e s p e e d a n d g r a d e a r e i n c r e a s e d e v e r y 3 m i n . Each p e r s o n continues e x e r c i s e until a s e l f - d e t e r m i n e d e n d p o i n t of e x h a u s t i n g f a t i g u e , m a r k e d
367
Kronmal and Tarter
d y s p n e a , aching o r w e a k n e s s in the l e g s , d i z z i n e s s , pain, o r v a r i o u s c o m b i n a t i o n s of t h e s e . . . .
chest
E a c h s u b j e c t w a s e x a m i n e d b e f o r e and a f t e r the t e s t by a p h y s i c i a n who r e m a i n e d in a t t e n d a n c e during the e n t i r e p r o cedure. . . . . H e a r t r a t e s w e r e r e c o r d e d at r e s t and during e a c h m i n u t e of e x e r c i s e and r e c o v e r y , while blood p r e s s u r e s w e r e obtained at r e s t and on i n i t i a l r e c o v e r y as w e l l a s 3 and 6 m i n . a f t e r e x e r t i o n . E x e r c i s e p e r f o r m a n c e w a s e v a l u a t e d b y the following p a r a m e t e r s of c a r d i a c function: (1) t o t a l d u r a t i o n of e x e r t i o n , (2) m a x i m a l h e a r t r a t e , and (3) m a x i m a l s y s t o l i c blood p r e s s u r e . . . . R e s t i n g h e a r t r a t e and blood p r e s s u r e w e r e obtained while e a c h s u b j e c t w a s sitting j u s t b e f o r e the s t a r t of the e x e r c i s e . The a i m of the study w a s p r i m a r i l y d e s c r i p t i v e and e x p l o r a t o r y . Thus the a u t h o r s w e r e i n t e r e s t e d in t e c h n i q u e s that would d e s c r i b e t h e i r d a t a w e l l and allow t h e m to d e v e l o p h y p o t h e s e s f o r f u t u r e s t u d i e s . The u s e of o r t h o g o n a l p o l y n o m i a l n o n p a r a m e t r i c d e n s i t y e s t i m a t i o n p r o v i d e d t h e m with a u s e f u l t o o l f o r t h e s e purposes. SOME E X A M P L E S O F T H E USE O F ORTHOGONAL P O L Y NOMIAL D E N S I T Y E S T I M A T E S Although t h e r e a r e m a n y u s e s of d e n s i t y e s t i m a t e s , p a r t i c u l a r l y in the a r e a of m u l t i v a r i a t e a n a l y s i s , the application d e s c r i b e d h e r e is r e s t r i c t e d t o the d e s c r i p t i o n of d a t a . The f o c u s i s on e s t i m a t i o n of the c u m u l a t i v e , the u n i v a r i a t e d e n s i t y , and b i v a r i a t e d e n s i t y ( T a r t e r and K r o n m a l , 1 9 7 0 ; K r o n m a l and T a r t e r , 1 9 6 8 ; and C e n c o v , 1 9 6 2 ) . We i l l u s t r a t e this a p p l i c a t i o n by e x a m p l e s f r o m the d a t a d e s c r i b e d e a r l i e r a s w e l l as with s o m e Monte C a r l o s i m u l a t i o n d a t a .
368
Density Estimates
The u s e of a n e w technique f o r the d e s c r i p t i o n of data m u s t be j u s t i f i e d on at l e a s t t h r e e grounds: i t s m a t h e m a t i c a l s t a t i s t i c a l p r o p e r t i e s , i t s a c c e p t a b i l i t y to applied s t a t i s t i c i a n s , and m o s t i m p o r t a n t l y i t s u s e f u l n e s s to the r e s e a r c h e r . Although t h e r e a r e a n u m b e r of p a p e r s on the m a t h e m a t i c a l a s p e c t s of o r t h o g o n a l p o l y n o m i a l e s t i m a t i o n , t h e r e i s l i t t l e to s u g g e s t the u s e of t h e s e m e t h o d s f o r d e s c r i p t i v e p u r p o s e s . H o w e v e r , this i s to be e x p e c t e d for a topic as new and as m a t h e m a t i c a l l y d i f f i c u l t as t h i s one b e c a u s e of its c o m p l e x i t y and the lack of a c l e a r d e f i n i t i o n as to what c o n s t i t u t e s a good d e s c r i p t i o n of the data. The l a r g e Monte Carlo s t u d i e s d e s c r i b e d b y A n d e r s o n (1969) s e e m to support the v i e w that for s a m p l e s i z e s as s m a l l a s 50, e x c e l l e n t e s t i m a t e s of the underlying p r o b a b i l i t y d e n s i t y a r e obtained through the u s e of the o r t h o g o n a l p o l y n o m i a l e s t i m a t e s . F i g u r e s 1 and 2 show the e s t i m a t e s of a d e n s i t y and c u m u l a t i v e r e s p e c t i v e l y f r o m a s a m p l e of s i z e 200 f r o m the n o r m a l d i s t r i bution with m e a n 0 and v a r i a n c e 1. F i g u r e 3 s h o w s a graph of the a p p r o x i m a t e p r o b a b i l i t y c o n t o u r s for a b i v a r i a t e n o r m a l with m e a n s e q u a l to z e r o and v a r i a n c e s equal to 1 and c o r r e l a t i o n equal to 0. 69 b a s e d on a s a m p l e s i z e of 100. The p r o g r a m g e n e r a t i n g the c o n t o u r s i s s t i l l in the d e v e l o p m e n t a l s t a g e , so t h e s e plots appear on a line printer and thus a r e s o m e w h a t d i s t o r t e d . Although t h i s p r o g r a m w a s u n a v a i l a b l e at the t i m e the K a s s e r - B r u c e study w a s a n a l y z e d , we u s e it to i l l u s t r a t e the potential v a l u e of g r a p h i c a l d i s p l a y s of b i v a r i a t e c o n t o u r s . One c a n s e e f r o m f i g u r e s 1, 2 and 3 that the e s t i m a t e s g e n e r a t e d by t h e s e m e t h o d s s e e m to be quite adequate for d e s c r i b i n g the populations f r o m w h i c h the data w e r e g e n e r a t e d . To our k n o w l e d g e , l i t t l e u s e h a s b e e n m a d e of any of the m e t h o d s f o r n o n p a r a m e t r i c d e n s i t y e s t i m a t i o n (with the p o s s i b l e e x c e p t i o n of i t s u s e in c l a s s i f i c a t i o n ) . In a s t a t i s t i c a l s y s t e m of c o m p u t e r p r o g r a m s (CCSS) for the m a i n t e n a n c e and d e s c r i p t i o n of c o m p l i c a t e d data f i l e s , a p r o g r a m i s included that p r o d u c e s e s t i m a t e s of the u n i v a r i a t e d e n s i t y and c u m u l a t i v e (Kronmal et al, 1970). Other than t h i s package of p r o g r a m s and its u s e by
369
Kronmal and Tarter
~4
-3
-2
F i g u r e 1. Estimate of density for sample of size 200 f r o m N(0, 1).
370
Density Estimates
F i g u r e 2 . E s t i m a t e of cumulative distribution for sample s i z e 2 0 0 f r o m N(0, 1).
371
Kronmal and Tarter
• • o o I* I/) < O UJ > -111 I O! Zl^-oi h- 00 oc u. id o o a u. • > u. >- I- X O K O —• V) w ^ V> Z Ui Uj Z UJ X ID UJ C) X O X3 uj 3 z >- X X t- KX • _J U. < t"OIO CO _J < t- ai a. CO X X QOhUi OC !-> X a. uj u. hXo u. z o x >- o Z> -J x uj r\j 0C > x S< H«-< t- x u O} Z uj < OUJH 4UJZ u. x a: o a»- x xMM UJ o in t£t < t C MC MC M C M C MC C MC M M •i MC M C MC M •J' C MC C MIM o »0 >c •4C MC M 0 >0 C MC M •O >C C M si" 00 CO00 CO 00 O C M IM C M ^ >0 to C O00 CD CajO w a>
376
Density Estimates
IM IM CM
UJ
OC
CM IM
CM CM CM CM CM IM IM IM IM z z 3O ZM -I tX < < > X et UJ ^ CO O CO o » o Z Of
111 3 K < Z M OC • < I/) > oc «•> 3 CO O I0. z o o o UJ K U. t -4# * M CM CM •4" * 1- -4- •4" * IM IM IM •4" -t -4- •4"•4 CM * •4- >0 •c «o 0 «O CMCM «t >0 «o * « « O > 0 S 3 Z Z Z • O C M I M •4« «O »O vO «o
CO
IM
•o
CD CO CO
rg CM
-o
eo
CM
•c
3 O K Z O U
oo
rg fM
X i
« Z a> o < «« CO t o < oe z o. < _i a a. UJ x
«t rg
>0
00
-o
oo 0
00 00
* *
* o * o # » * tri • CO
rg
oo
X
*
rg
0
CM rg rg -i-
*
* *
rg
-o
CO 00
00 CD
*
•
IM -i-
00 00
o >0
rg rg
co oo oo oo
rg rg rg o-c- op CO oo00 00 00 CO 00 00 00 LL Z a.
„o „ HH o o oo o
•
*
CT"
381
Kronmal and Tarter o + o• i in o f\J
•m
•
m H » oo M iz I/) tM JO a a. m x u. >- o i/i oe UJ UJ Xm t- x 3
*
a ert
m • 00 II o 00 i
+
< —
+
X at i/i O UJ z t< UJ z X M KQ OC (to o o u. u
o (Pi • 01 ^
•
+
o o • m >0
>
•o o o
t—
UJ UJ ac x a. tO >• O -J O uj _i Ha> < X o « i- X -i a o or i- a. < I00 z u. a
• a
+
+
•
o m •
00
a
o
00 I X
m CM -Io
II 0c
*
(M
*
m
»
•o -i
i . e . , w e u s e e x p r e s s i o n (16) f o r o u r i n c l u s i o n r u l e . A
It i s w o r t h n o t i n g t h a t f ^ w i l l b e a r e a l v a l u e d f u n c t i o n if t h e t e r m s e l e c t i o n r u l e g i v e n in f o r m u l a (16) i s u s e d s i n c e if B^. i s selected then w i l l a l s o be s e l e c t e d and t h i s h a s t h e e f f e c t of c a n c e l l i n g o u t t h e i m a g i n a r y p o r t i o n of f j ^ . CONCLUSION We w o u l d l i k e to e m p h a s i z e t h a t w e p r o f e s s no s p e c i a l i n s i g h t i n t o t h e u s e f u l n e s s o r v a l u e of t h e s e m e t h o d s . We do f e e l , h o w e v e r , t h a t s u c h m e t h o d s m a y f i n d a p l a c e in t h e t o o l s a v a i l a b l e t o t h e s t a t i s t i c i a n and i n v e s t i g a t o r f o r t h e d e s c r i p t i o n of d a t a . ACKNOWLEDGEMENT T h a n k s t o M r . S. R a m a n , a c a n d i d a t e f o r a P h . D. in B i o s t a t i s t i c s a t B e r k e l e y , f o r t h e u s e of h i s p r o g r a m f o r t h e c o n t o u r s e s t i m a t i o n . We w o u l d a l s o l i k e t o e x p r e s s o u r a p p r e c i a t i o n t o D r s . B r u c e and K a s s e r f o r p e r m i s s i o n to r e p r i n t p o r t i o n s of t h e i r p a p e r and f o r the u s e of t h e i r d a t a . This r e s e a r c h w a s s u p p o r t e d in p a r t by PHS R e s e a r c h C a r e e r Award Grant 1 -K03-GM38645-01.
387
Kronmal and Tarter DATA SET DESCRIPTION
DATA - KASSER AND BRUCE CORONARY HEART DISEASE STUDY (R. KRONMAL) CASES AND VARIABLES 234 CASES (SUBJECTS) 34 VARIABLES - 16 MEASUREMENT, 16 CATEGORICAL, 2 IDENTIFICATION BRIEF DESCRIPTION VAR
NAME
***
****
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
SUBJECT NUMBER CARD NUMBER SOURCE SEX AGE FUNCTIONAL CLASS ACTIVE DIGITALIS TREATMENT 12-LEAD ECG RESULTS RHYTHM SYSTOLIC BLOOD PRESSURE RESTING HEART RATE MAXIMUM HEART RATE DURATION ON TREADMILL SYSTOLIC BLOOD PRESSURE AT PHYSICAL EXAM DIASTOLIC BLOOD PRESSURE AT PHYSICAL EXAM MAXIMUM SYSTOLIC BLOOD PRESSURE ON TREADMILL MAXIMUM DIASTOLIC BLOOD PRESSURE ON TREADMILL HISTORY OF MYOCARDIAL INFARCTION HISTORY OF ANGINA PECTORIS ECG XRAY HISTORY OF HIGH BLOOD PRESSURE SURVIVAL STATUS MONTHS NUMBER OF PAST MYOCARDIAL INFARCTIONS ANGINA PECTORIS CONGESTIVE HEART HEART RATE AT STAGE ONE ON TREADMILL CHANGE IN HEART RATE SYSTOLIC BLOOD PRESSURE TIMES HEART RATE VOLUME OF OXYGEN PREDICTED FOR HEALTHY MAN WITH SIMILAR CHARACTERISTICS ACTUAL ESTIMATED VOLUME OF OXYGEN FOR THIS MAN FUNCTIONAL AEROBIC IMPAIRMENT
PURPOSE OF STUDY THE PURPOSE OF THE STUDY IS TO EXPLORE THE RELATIVE CONTRIBUTIONS OF AGE AND CORONARY HEART DISEASE TO THE IMPAIRMENT OF FUNCTIONAL CAPACITY. THE SAMPLE CONSISTS OF 117 MALE CORONARY PATIENTS AND 117 AGE-MATCHED HEALTHY MIDDLE-AGED MEN. REFERENCE KASSER, IRWIN S. AND ROBERT A. BRUCE. COMPARATIVE EFFECTS OF AGING AND CORONARY HEAiRT DISEASE AND SUBMAXIMAL AND MAXIMAL EXERCISE. CIRCULATION 39, 1969, 759-774.
388
Density Estimates DETAILED DESCRIPTION VAR
COL
FORMAT
DESCRIPTION
***
***
******
***********
1
1-3
XXX.
2
4
X.
3
5
X.
4
6
X.
5 6
7-8 9
XX. X.
7
10
X.
8
11
X.
9
12
X.
10
13
X.
11
14-16
XXX.
12 13 14
17-19 20-22 23-25
XXX. XXX. XXX.
15 16
26-28 29-31
XXX. XXX.
17
32-34
XXX.
18
35-37
XXX.
19
38
X.
20
39
X.
21
40
X.
SUBJECT NUMBER 001 - 364 = CONTROL GROUP 401 - 517 = CORONARY GROUP CARD NUMBER 1 = CARD 1 SOURCE 1 = CLINIC 5 = FACULTY 7 = YMCA SEX 1 = MALE AGE (YEARS) FUNCTIONAL CLASS 0 = NONE 1 = MINIMAL 2 = MODERATE 3 = MODERATE TO SEVERE 4 = SEVERE ACTIVE 0 = UNKNOWN 1 = VERY 2 = NORMAL 3 = LIMITED DIGITALIS TREATMENT 0 = NO 1 = YES 12-LEAD ECG RESULTS 0 = UNKNOWN 1 => NORMAL 3 = ABNORMAL RHYTHM 0 = UNKNOWN 1 = NORMAL 2 = ATRIAL FIBRILLATION 3 = OTHER SYSTOLIC BLOOD PRESSURE 0 = UNKNOWN RESTING HEART RATE MAXIMUM HEART RATE DURATION ON TREADMILL (SECONDS) 999 = UNKNOWN SYSTOLIC BLOOD PRESSURE AT PHYSICAL EXAM DIASTOLIC BLOOD PRESSURE AT PHYSICAL EXAM 999 = UNKNOWN MAXIMUM SYSTOLIC BLOOO PRESSURE ON TREADMILL 999 = UNKNOWN MAXIMUM DIASTOLIC BLOOD PRESSURE ON TREADMILI 999 = UNKNOWN HISTORY OF MYOCARDIAL INFARCTION 0 = NONE 1 = POSSIBLE 2 = DEFINITE 9 = CONTROL GROUP HISTORY OF ANGINA PECTORIS 0 = NONE 1 = POSSIBLE 2 = DEFINITE 9 = CONTROL GROUP ECG
389
Kronmal and Tarter
22
41
X.
23
42
X.
24
43
X.
25
44--45
XX.
26
46
X.
27
47
X.
28
48
X.
29
49--51
XXX.
30
52--54
XXX.
31
55--57
XXX.
32
58--60
XXX.
33
61--63
XXX.
34
64--66
XXX.
FORMAT IS
0 = NORMAL 1 = POSSIBLE MYOCARDIAL INFARCTION 2 = PROBABLE MYOCARDIAL INFARCTION 3 = OTHER 9 = UNKNOWN OR CONTROL GROUP XRAY 0 = NORMAL 1 = ENLARGED 9 = UNKNOWN OR CONTROL GROUP HISTORY OF HIGH BLOOD PRESSURE 0 = NORMAL 1 = HYPERTENSIVE 9 = CONTROL GROUP SURVIVAL STATUS I = ALIVE 2 = DEAD 8 = CONTROL GROUP 9 = UNKNOWN MONTHS 99 = UNKNOWN OR CONTROL GROUP NUMBER OF PAST MYOCARDIAL INFARCTIONS 1 . ] 1 — I 2 = 2 3 = NONE 8 = CONTROL GROUP 9 = UNKNOWN ANGINA PECTORIS 1 = SLIGHT 2 = SIGNIFICANT 3 = NONE 8 = CONTROL GROUP 9 = UNKNOWN CONGESTIVE HEART I = YES 3 = NONE 8 * CONTROL GROUP 9 = UNKNOWN HEART RATE AT STAGE ONE ON TREADMILL 999 = UNKNOWN CHANGE IN HEART RATE (MAXIMUM HEART RATE RESTING HEART RATE) SYSTOLIC BLOOD PRESSURE TIMES HEART RATE 999 = UNKNOWN VOLUME OF OXYGEN PREDICTED FOR HEALTHY MAN WITH SIMILAR CHARACTERISTICS ACTUAL ESTIMATED VOLUME OF OXYGEN FOR THIS MAN 999 = UNKNOWN FUNCTIONAL AEROBIC IMPAIRMENT 999 = UNKNOWN
(F3.0t3F1.0»F2.0,5F1.0,8F3.0,6F1.0,F2.0,3F1.0,6F3.0)
N = 23* LOCATION OF DATA CARD IMAGE - FS.C073.CKASS1 LISTING OF DATA (THE FIRST TWO LINES ARE COLUMN GUIDES» 0 1 2 3 4 5 6 12345 6789 0123456789 012345678 901234567 89012 345678 901 234 567 890 123 456 117 1480 1031 80 76 180577120 217 1490 1011 80 68 176785120
90200 80 99999 899888 114 104 359 388 359 7 80200 80 99999 899888 95 108 351 385 461 - 1 9
390
Density Estimates 317 417 617 1417 1617 2117 2517 2917 3017 3117 3317 3417 3617 3817 4417 4517 4817 4917 5117 5317 5517 5617 5717 5917 6017 6217 6617 6717 6817 7017 7117 7217 7317 7417 7917 8217 8417 9017 9217 9417 10317 10517 10817 10917 11117 11717 11917 12017 12217 12517 13117 13217 13417 13617 13717 14117 14917 15217 15317 15517 15717 16317 16817 17417 17617 17917 18017
1410 1520 1520 1480 1450 1510 1590 1440 1590 1440 1620 1420 1520 1460 1510 1460 1660 1390 1520 1450 1540 1540 1490 1560 1450 1600 1480 1500 1600 1410 1600 1520 1480 1550 1590 1390 1720 1550 1390 1540 1520 1560 1460 1390 1380 1610 1460 1490 1530 1380 1600 1490 1500 1510 1570 1390 1520 1480 1530 1530 1480 1420 1520 1610 1^30 1630 1470
1011 4 8 70 1031 46 57 1011 40 66 1011 10 6 0 2011 38 84 2031 84 68 2011 70 64 1 0 1 1 1 8 0 72 1011 9 4 75 1011 0 57 1011 78 76 1011 36 63 1011 76 74 2011 80 65 1011 6 8 54 1011 40 62 1011 60 70 1 0 1 1 1 4 0 64 1011 50 45 1011 70 66 1011 0 68 1011160 67 1011 70 60 1011110 68 1011 6 4 55 1011 56 84 1011 26 66 1031 90 58 1011 30 77 1011 80 63 1011 46 65 1011110 60 1011 54 80 1031 56 78 1031 38 68 2011 50 65 1 0 3 1 3 0 SO 2031 76 88 1031 20 78 1011 88 64 1011 30 56 1011 2 4 64 1011 84 80 1011 40 66 1011 20 68 1011 62 88 2011 50 80 2011 2 2 54 2011 62106 2011 58 56 2011 48 75 2033 82 64 2011 76 80 2 0 1 1 2 0 72 2011 6 0 88 2031 6 0 86 2011 20 62 1011 62 76 2011 60 76 2 0 1 1 1 0 2 80 2011 3 4 75 1 0 1 1 5 0 56 2011 4 0 80 1011 10 68 2011 0 84 1011 52 58 1 0 1 1 2 6 72
174775132 170624144 170567110 174720130 190570112 182517 96 148558120 194607110 180583116 184765190 152523112 190688146 152496104 174445120 150635122 170630130 156476150 194585140 158705140 170675150 184695150 190740120 152523150 176675130 160555 96 184555130 174676114 180750130 174602110 180708110 180590144 166624130 166681156 170647120 184578112 180575110 130218170 182573136 180610130 184660138 156640130 168580130 176560126 180742140 170623120 150262118 190580130 182585118 190596148 184575122 160425112 168550108 190575164 190610130 200455140 200610120 184618110 176540118 184615140 184624118 184583106 176 790130 170443100 148487130 194575130 156465134 180551124
8 2 1 8 0 60 9 9 9 9 9 8 9 9 8 8 8 92190 60 99999 899888 68150 60 99999 899888 9 0 1 2 0 60 9 9 9 9 9 8 9 9 8 8 8 82150 60 999 99 899888 62180 30 99999 8 9 9 8 8 8 8 0 1 9 0 50 9 9 9 9 9 8 9 9 8 8 8 70290 60 99999 899888 8 8 2 1 0 80 99999 899888 90190 60 99999 899888 66190 80 9 9 9 9 9 899888 96110 60 99999 899888 84180 80 99999 899888 80200 70 99999 899888 7 0 1 9 0 70 9 9 9 9 9 8 9 9 8 8 8 8 0 1 7 0 70 9 9 9 9 9 8 9 9 8 8 8 76210 60 99999 899888 80280 60 99999 899888 9 0 1 9 0 70 9 9 9 9 9 8 9 9 8 8 8 70220 80 99999 899888 9 0 2 2 0 80 9 9 9 9 9 8 9 9 8 8 8 80280999 99999 899888 80 2 2 0 80 9 9 9 9 9 8 9 9 8 8 8 8 0 2 4 0 60 9 9 9 9 9 8 9 9 8 8 8 80160 70 9 9 9 9 9 8 9 9 8 8 8 8 0 1 8 6 70 9 9 9 9 9 8 9 9 8 8 8 8 0 1 4 0 80 9 9 9 9 9 8 9 9 8 8 8 100220100 99999 899888 8 0 1 4 0 50 9 9 9 9 9 899888 9 0 1 9 0 70 9 9 9 9 9 8 9 9 8 8 8 100190 80 99999 899888 9 0 2 4 0 50 9 9 9 9 9 8 9 9 8 8 8 100210 90 99999 899888 78176 60 9 9 9 9 9 899888 78150 60 99999 899888 7 6 1 6 0 50 9 9 9 9 9 8 9 9 8 8 8 90200 90 99999 899888 8 2 2 1 2 80 9 9 9 9 9 8 9 9 8 8 8 8 0 1 5 0 70 9 9 9 9 9 8 9 9 8 8 8 1 0 0 2 3 6 90 9 9 9 9 9 899888 8 0 1 6 0 50 9 9 9 9 9 8 9 9 8 8 8 9 0 1 5 4 80 9 9 9 9 9 8 9 9 8 8 8 8 0 2 1 0 40 9 9 9 9 9 8 9 9 8 8 8 90180999 99999 899888 88140 70 99999 899888 82180 80 99999 899888 80180 60 99999 899888 82140 70 99999 899888 94210 96 99999 899888 9 0 1 8 0 80 9 9 9 9 9 899888 8 0 1 6 0 60 9 9 9 9 9 899888 74190 60 99999 899888 9 0 2 4 0 80 9 9 9 9 9 899888 100150 90 99999 899888 8 6 2 0 0 80 9 9 9 9 9 899888 90180 92 99999 899888 8 0 1 3 0 70 9 9 9 9 9 8 9 9 8 8 8 90180 60 99999 899888 80200 70 99999 899888 76220999 99999 899888 90140 60 99999 899888 84180 80 999 99 899888 8 0 1 4 0 80 9 9 9 9 9 8 9 9 8 8 8 9 0 1 2 0 70 9 9 9 9 9 8 9 9 8 8 8 8 0 1 3 0 70 9 9 9 9 9 899888 100186 60 9 9 9 9 9 899888 80150 40 99999 899888
391
91 98 96 96 126 106 88 106 106 106 126 102 100 106 86 95 116 110 70 96 108 108 90 94 104 118 108 85 106 90 100 90 116 100 112 106 120 114 118 100 86 92 108 95 100 138 112 92 130 108 114 106 116 102 132 136 102 128 9 99 120 100 96 122 999 124 106 104
104 113 104 114 106 114 84 122 105 127 76 127 78 109 96 108 86 130 113 104 116 123 92 108 105 100 108 12 2 97 117 115 106 86 92 116 115 50 94 102 120 100 104 96 114 102 62 H O 128 84 128 85 104 110 118 112 114 122 100 108 104 109 120 90 80 110 98 108
313 322 254 208 284 327 281 562 377 349 288 208 273 34 7 284 288 32 7 543 300 373 404 531 334 422 255 342 243 395 243 341 341 398 348 299 275 287 259 385 269 434 249 258 369 323 237 269 341 254 398 331 255 319 455 284 399 359 239 316 367 404 257 316 237 177 252 290 269
407 456 377 382 377 354 388 429 365 355 349 329 328 349 399 373 359 362 399 451 351 332 404 413 377 319 362 294 380 387 393 385 340 309 412 363 377 421 396 40 7 372 417 372 439 385 332 367 407 396 348 356 348 388 407 383 443 356 371 4 0 7 42 3 356 365 377 382 388 410 369 393 359 359 381 358 324 183 338 357 412 375 372 399 377 390 367 360 393 350 412 440 415 381 353 204 362 360 354 363 344 368 384 358 325 284 354 345 352 358 349 375 333 299 381 375 346 379 388 341 344 377 344 382 357 362 4 0 4 46 3 346 293 353 315 344 358 348 304 391 346
-12 -1 6 -10 2 5 -6 6 0 -13 5 -2 15 18 -1 2 8 12 -11 -2 -11 -17 13 -10 12 2 -4 -15 -4 -3 -2 -1 -5 -6 0 6 43 -5 9 -7 -3 1 10 -6 8 42 0 -2 -7 6 12 2 -1 -7 10 1 -9 12 -9 -11 -1 -14 15 10 -4 12 11
Kronmal and Tarter 18417
1480
2011
30
78
180545120
72150
50
99999
899888
100
102
269
357
343
18817
1520
1011
20
56
184605140
88160
60
9 9 9 9 9
899888
103
128
294
377
372
1
19017
1400
2011
80
64
164450118
84198
60
99999
899888
100
100
324
378
296
21
3
19117
1480
1011
52
62
176650118
80170
60
99999
899888
104
114
299
388
394
- 1
19317
1540
1011
76
62
184
90210
80
99999
899888
102
122
386
372
429
- 1 5
19517
1390
2011
32
62
184630110
70142
50
99999
899888
94
122
261
381
385
0
19817
1450
1011
54
64
190710136
66190
70
99999
899888
110
126
360
396
424
- 7
720134
19917
1570
2011
18
80
130245124
80142
70
9 9 9 9 9
899888
999
50
184
333
196
41
20017
1480
1011
40
60
164605160
80200
60
99999
899888
94
104
327
388
372
4
20117
1420
1011
36
74
186595100
80136
60
99999
899888
114
112
252
404
368
9
20317
1390
2011
28
75
146450112
80140
50
9 9 9 9 9
899888
999
71
204
381
296
22
20617
1490
1011
80
72
1 7 4 4 7 8 1 5 0
70230
50
99999
899888
102
102
400
385
310
19
20717
1720
1031
82
65
1 6 0 3 6 0 1 2 4
82206
90
9 9 9 9 9
899888
156
95
329
324
252
22
20917
1390
1011110
74
1 9 4 7 6 3 1 1 0
90220
60
9 9 9 9 9
899888
98
120
426
412
450
- 9
21017
1440
2011
3 0
72
1 8 0 5 5 0 1 1 0
80140
40
99999
899888
100
108
251
368
345
6
21717
1391
1011
9 4
72
176598156
100250
60
99999
899888
104
104
439
412
369
10 4
21817
1430
1011
6 0
84
178628130
90190
60
99999
899888
114
94
338
4 0 1
384
22117
1540
1011
90
80
1 7 4 5 9 3 1 7 0
110260
60
99999
899888
96
94
452
372
36 7
2 2 4 1 7
1550
1011
60
58
1 8 0 6 6 0 1 2 0
80180
80
9 9 9 9 9
899888
94
122
323
369
399
- 8 2
1
22517
1420
2011
20
68
184590110
84130
70
9 9 9 9 9
899888
110
116
239
373
365
30115
1610
2031
10
68
1 6 0 4 6 5 1 1 4
90105
80
99999
899888
108
92
167
322
304
5
30715
1510
2011
66
86
164414130
88196
60
9 9 9 9 9
899888
114
78
321
349
279
20 - 7
31315
1630
1011
46
55
180610130
80176
70
9 9 9 9 9
899888
96
125
316
348
375
31715
1610
2011
30
72
158401130
80160
90
99999
899888
106
86
252
322
272
15
31915
1520
2001
38
66
172564122
9 0 1 6 0
82
9 9 9 9 9
899888
102
106
275
346
352
- 1 - 4
32015
1550
2011
90
68
164568120
84210
70
9 9 9 9 9
8 9 9 8 8 8
74
96
344
338
354
32315
1710
2011
24
61
160360120
84144
60
9 9 9 9 9
899888
122
99
230
29 5
252
14
32415
1380
2011
3 0
88
2 0 0 6 3 7 1 5 0
84140
70
99999
899888
150
112
279
384
388
- 1
32615
1630
2031
20
74
170405130
94150
80
99999
899888
116
96
254
317
274
13
32715
1640
2001
48
68
1 4 0 3 5 7 1 2 0
80168
72
9 9 9 9 9
899888
105
72
235
314
251
20
32815
1500
2011
80
68
2 0 0 6 0 0 1 1 0
8 4 1 9 0 1 0 0
9 9 9 9 9
899888
114
132
379
352
370
- 5
3 2 9 1 5
1580
2031
46
74
166530140
90186
80
9 9 9 9 9
899888
124
92
308
330
336
- 1 18
33315
1580
1011
62
66
1 7 2 4 4 8 1 1 0
70172
70
9 9 9 9 9
899888
112
106
295
361
2 9 6
33415
1380
0001
88
65
174477112
68200
70
9 9 9 9 9
899888
999
109
347
384
310
19
33715
1390
2032
66
86
190646140
9 9 9 2 0 6
80
9 9 9 9 9
899888
126
104
391
381
393
- 2
33815
1570
2011
40
58
1 6 8 5 9 6 1 2 0
80160
80
9 9 9 9 9
899888
104
110
268
333
368
- 1 0
33915
1500
1011
54
76
1 8 0 5 0 0 1 5 0
110210
90
9 9 9 9 9
899888
130
104
377
383
321
16
34815
1610
2011
38
84
188508142
100180
80
99999
899888
134
104
338
322
325
35215
1680
2033
40
56
1 7 5 5 7 7 1 4 0
80180
80
99999
899888
100
119
314
303
359
35315
1410
1011
4 4
74
174605
70140
60
9 9 9 9 9
899888
102
100
243
407
372
8
35415
1550
1011
50
76
168497130
80180
90
99999
899888
104
92
302
369
320
13
35515
1500
2011
12
76
180512118
84130
40
9 9 9 9 9
899888
999
104
233
352
32 7
35615
1540
2011
44
82
1 8 0 5 5 0 1 2 6
86170
70
99999
899888
114
98
305
341
345
96
0 - 1 8
7 - 1
3 5 7 1 5
1480
1011
66
63
1 8 0 5 9 0 1 2 4
80190
60
9 9 9 9 9
899888
110
117
341
388
365
5
36015
1 5 5 0
2031
40
70
152438120
76160
70
9 9 9 9 9
899888
106
82
243
338
291
14
36115
1620
2011
74
72
154510146
76250
78
99999
899888
104
82
384
320
326
- 1
36315
1710
2001
30
60
1 6 0 4 5 0 1 6 0
90190
70
99999
899888
114
100
303
295
296
0
3 6 4 1 5
1480
2031
50
85
180637110
80160
90
9 9 9 9 9
899888
112
95
287
357
388
- 8
40111
1422
2031
56
88
166545104
70160
80
12200
141123
122
78
265
373
343
8
40211
1662
2001
56
90
156345150
9 0 1 9 0
90
12200
9 9 9 9 9 9
122
66
296
309
245
20 64
40311
1562
0000
9
90
153
80130
80
22000
212199
153
63
198
336
120
40411
1552
2001
20
82
1 3 0 4 1 0 1 0 0
80128
80
2 2 9 0 0
9 9 9 9 9 9
110
48
166
338
277
18
40511
1412
2001
30
84
160385170
120210
90
12211
145999
135
76
335
376
265
29
9 0 1 0 4
40611
1620
0000
50
82
160240150
90200
80
20201
148333
152
78
319
320
194
39
40711
1462
2001
40
75
130415150
100190
98
12001
154313
130
55
246
362
279
22
40811
1442
1001
0
83
1 3 5 7 1 0 1 5 0
90190
85
02001
2
112
52
256
399
424
- 6
40911
1501
2031
40
55
1 6 0 5 6 5 1 4 0
9 0 1 8 0 : LOO
02201
223131
78
105
287
352
353
0
41011
1733
3031
10
92
140135120
75130
75
02210
221199
135
48
181
290
142
50 33
9123
41111
1482
2031
10
65
145330130
85140
85
22200
124929
110
80
202
357
238
41211
1532
2001
10
56
120275130
80140
80
22210
2 2 9 3 1 3
102
64
167
344
211
38
41311
1513
1031
18
70
1 1 8 3 1 5 1 0 4
80140
85
12301
2 2 0 1 2 3
100
48
165
380
230
39
125
72
346
328
267
18
97
24
239
341
207
39
41411
1590
0000
60
82
154390165
1 0 0 2 2 5 1 2 5
02311
146331
41511
1543
3031
45
76
1 0 0 2 6 8 1 9 5
105240120
12301
2
8129
41611
1412
2031
58
61
164565130
9 0 1 8 8 1 0 5
21201
122313
108
103
308
376
353
6
41711
1562
2031
0
76
1 3 0 1 0 0 1 3 0
100999999
20211
2
130
54
999
336
.125
62
41811
1380
2001
50
70
162710150
90200110
02001
148323
97
92
323
384
424
- 1 0
41911
1403
3031
20
85
1 6 0 2 5 0 1 1 0
70130
22200
1
138
75
207
378
198
4 7
90
392
4199 8313
Density Estimates 42011 42111 42211 42311 42411 42511 42611 42711 42811 42911 43011 43111 43211 43311 43411 43511 43 6 1 1 43711 43811 43911 44011 44111 44211 44311 44411 44511 44611 44711 44811 44911 45011 45111 45211 45311 45411 45511 45611 45711 45811 45911 46011 46111 46211 46311 46411 46511 46611 46711 46811 46911 47011 47111 47211 47311 47411 47511 47611 47711 47811 47911 48011 48111 48211 48311 48411 48511 48611
1421 151 1 1521 13 70 1481 1350 1351 1483 1522 1462 1513 1500 1723 1563 1563 1632 1531 1530 1573 1571 1622 1732 1442 1633 1591 1511 1523 1640 1532 1581 1530 1582 1451 1423 1602 1341 1642 1351 1422 1532 1581 1381 1352 1342 1683 1493 1552 1580 1432 1392 1663 1502 1453 1530 1564 1492 1490 1562 1380 1390 1622 1703 1532 1682 1502 1462 1583
2 0 3 1 74 82 2 0 1 1 6 0 75 0001 0 73 1 0 1 1 6 0 56 2131 10102 0 0 0 0 3 0 75 1 0 3 1 1 1 0 98 3 0 3 1 10 84 2 0 1 1 4 8 74 3011 0 70 0001 12107 0 0 0 0 10 58 3011 35106 3031 35 65 3 0 3 1 2 5 56 2 0 1 1 4 5 77 2011 45104 1 0 0 0 5 8 76 2031 75 65 2 0 3 1 2 0 56 2 0 3 1 3 4 85 2 0 3 1 1 5 2 60 0 0 0 0 14 68 2 0 0 1 2 0 75 1001 20 68 0 0 0 0 44 62 0001 20 72 0 0 0 0 4 0 98 2031 40 70 2 0 1 1 2 4 96 2 0 3 1 4 0 75 2 0 1 1 3 8 52 2 0 3 1 7 0 82 2 0 1 1 4 5 78 2 0 3 1 15 72 0001 50 80 2000 30 58 2 0 3 1 4 0 88 2 0 1 1 35 76 2 0 3 1 6 4 87 2 0 0 0 5 0 97 2031 30 84 2031 14 90 3011 4 0 84 2 0 0 1 2 0 72 2 0 1 1 4 0 72 2031 40 70 2 0 0 0 27 70 2 0 1 1 12 6 3 1011 25 66 3031 0 61 2031 24 66 3 0 0 1 2 7 80 0000 0 78 3 0 3 1 2 5 78 2 0 3 1 1 5 75 0 0 0 0 30 93 2001 0 82 0 0 0 0 26 77 0 0 0 0 1 0 88 2 0 0 0 3 0 56 2 0 0 1 2 0 95 0000 30112 2 0 1 1 25 53 0 0 3 1 20 80 2031 45 70 2 0 3 1 2 8 82
170400110 175510140 173999100 190675130 172305130 130430130 180628140 160360105 166470122 150130150 141135168 125405150 126 3 0 1 7 0 148420125 114140105 162390130 146235105 132390100 120260125 132265190 154270180 104204152 128280116 134180140 110270140 150410116 122 85140 156475154 160421150 162590116 156420160 105135140 150412150 120110105 138213115 200420130 94210110 174445150 128140140 154 540136 152300160 134510110 140235128 152395120 100110130 164385120 120285150 120360118 112150140 118305115 122175200 134222120 175310140 128540118 124280205 154360165 168575140 165290160 120250134 120160130 115 7 5 1 9 0 98210120 152220130 110285135 138208160 136240115 130240100
80184 90 8 0 2 0 0 90 90999999 8 0 1 6 0 80 8 8 1 4 0 70 7 0 1 6 0 90 95250100 70130 84 85170 95 88190 90 98178100 8 0 1 7 0 80 8 5 2 0 5 90 7 8 1 6 0 90 6 0 1 3 0 80 9 0 1 7 5 90 65150 60 7 0 1 5 8 86 80200 80 7 8 2 1 0 85 9 0 2 1 4 96 88186104 72130 70 8 5 1 6 0 80 8 0 1 6 0 80 7 8 1 6 0 70 90160110 8 0 1 9 4 80 9 0 1 9 0 80 7 4 1 4 0 84 1 1 0 2 0 0 94 8 0 1 7 8 85 90220110 6 0 1 5 0 90 70160110 80180100 7 5 1 2 0 80 1 0 0 1 9 0 80 9 0 1 7 5 80 78200 40 8 8 2 1 0 85 8 0 1 4 0 80 90142 92 90160 90 6 0 1 5 0 80 70160 70 9 5 1 9 0 95 8 2 1 4 5 80 8 2 1 5 2 75 8 5 1 4 0 90 100220100 7 5 1 5 0 80 9 0 1 4 0 90 75170 70 110230120 122195152 9 0 1 7 0 90 100200100 8 6 1 6 0 80 90140100 1 0 0 1 6 0 90 8 6 1 4 0 70 8 0 1 6 0 90 7 5 1 6 0 80 96180100 80160 95 5 2 1 2 8 50
393
12200 02000 21210 02000 20210 12210 20200 02001 02001 02001 02300 02990 01201 22200 22300 12200 22010 20200 02210 02001 11200 02390 02000 22200 02000 20210 22200 10200 20200 02000 22201 02100 22001 02000 02211 10201 22000 20201 02100 20210 20211 01210 22200 22000 12300 02000 02301 22310 22000 02300 02211 22200 01200 02000 22201 22211 20200 12000 12010 22000 12100 22211 12301 22300 00211 0 2 2 00 22010
111323 111313 171313 1 5313 1 5333 141323 999999 1 8323 119323 1 6999 172313 156313 112333 139323 114323 120323 999999 157333 118121 120333 142123 999999 999999 999999 153313 161333 166323 151333 999999 999999 123313 122313 999999 130999 210199 999999 146333 123333 124323 122333 138333 999999 119323 124333 143313 117323 126323 242333 123313 129323 122323 131323 116323 143313 1 8323 128311 153333 168333 999999 999999 230121 252113 230199 122323 126333 126999 136323
144 1 16 113 97 150 108 134 144 100 140 141 125 116 98 114 116 134 100 114 104 999 100 118 128 98 88 122 124 115 116 118 103 107 120 136 138 90 120 128 128 140 98 122 115 100 128 120 104 112 114 116 116 170 120 116 114 117 120 105 120 100 98 142 90 128 128 112
88 100 100 134 70 55 82 76 92 80 34 67 20 83 58 85 42 56 55 76 69 44 60 59 42 88 50 58 90 66 81 53 68 42 66 120 36 86 52 67 55 50 50 68 28 92 50 50 49 52 61 68 95 50 46 79 75 83 43 32 59 3 40 57 58 66 48
312 349 999 303 240 207 449 207 282 284 250 212 258 236 148 283 218 208 239 277 329 193 166 214 175 239 195 302 303 226 311 186 329 179 220 359 112 330 223 307 319 187 198 243 149 262 227 173 170 165 268 200 244 217 285 300 285 329 191 167 183 137 243 175 248 217 166
373 349 346 418 357 392 423 357 346 362 349 352 293 336 336 317 344 375 333 333 320 290 368 317 359 349 346 314 344 330 344 330 365 373 325 395 314 392 373 344 330 384 392 395 303 354 338 330 370 412 309 352 365 344 336 354 354 336 384 381 320 298 344 303 352 362 330
27 272 326 6 999 999 407 2 36 225 287 26 384 9 29 252 306 11 140 61 59 142 274 21 91 68 16 282 145 56 267 15 191 44 28 267 38 203 206 38 34 208 176 39 41 213 48 164 41 208 277 20 118 65 309 1 17 282 365 - 1 0 282 17 56 142 23 278 65 130 180 44 28 282 179 42 294 24 61 145 341 0 223 32 326 15 191 51 270 31 130 57 265 25 216 36 23 252 149 59 225 45 162 47 185 47 37 228 341 0 213 36 252 28 358 0 218 34 198 48 154 59 113 64 179 39 184 46 216 28 178 49 194 46 41 194
Kronmal and Tarter 48711 48811 48911 49011 49111 49211 49311 49411 49511 49611 49711 49811 49911 50011 50111 50211 50311 50411 50511 50611 50711 50811 50911 51011 51111 51211 51311 51411 51511 51611 51711
1572 1553 1520 1612 1452 1512 1553 1511 1461 1691 1513 1491 1583 1383 1501 1381 1581 1690 1660 1492 1620 1440 1583 1452 1583 1542 1552 1682 1682 1471 1550
2031 3031 0001 2101 2031 0101 3031 0001 0001 2011 3031 1001 3011 3131 1031 3031 0001 0001 0000 2011 0000 0001 0101 2031 3031 2001 2001 2001 2031 2031 0000
50 62 30 80 40 75 52 68 50 86 20 85 40 70 10 66 20 90 55 60 20 88 70 77 18 76 30 75 90 76 15 98 30 87 20 75 10 70 52 64 4100 20100 40 78 55 77 5 77 20 88 40 88 20 64 45 60 26 75 40 84
95 154558120 92 261 76170 68 00200 110313 40 173 120120115 75145 65 20200 110333 118 50 212 125150130 85170 90 12300 231299 110 173610118 70170 70 00200 212311 125 105 294 56 283 142243150 100200100 02000 127323 126 69 200 154355110 70130 90 22210 2 8111 140 140540140 94180 90 21201 144213 103 70 251 168295150 90170 90 02200 999999 120 102 285 90160 80 22200 999999 118 80 271 170360140 56 220 116240135 70190 70 02000 128999 114 68 249 156240140 100160100 22211 124123 136 96 363 173640140 90210 94 01201 125333 128 64 173 60124 70 12000 140913 115 140240102 160570120 88150 95 222 00 124323 116 85 239 174580140 100230 90 20000 121333 124 98 400 62 239 160 65135 75150 80 22200 129323 160 160360140 88999999 20200 172333 125 73 999 110 80170 35 208 90190110 02110 165333 102 44 170 114240160 90150 90 02000 212999 110 66170 60 02010 999999 98 43 181 107180118 50 164 150195116 90110 96 22000 218999 142 72 309 172215160 95180100 02001 159333 142 64 255 142165140 80180100 22210 229111 124 150390115 80170 80 12200 133313 9 99 73 254 134100165 90170 90 22211 128323 126 57 227 175 60130 90150100 12200 157323 110 87 262 150260126 90166 90 21211 999999 116 62 248 51 172 115120130 80150 90 12000 999999 110 60 221 120315140 85185 70 22210 129999 100 180745104 68130 70 22201 134333 110 105 233 56 237 140115130 70170 70 02000 145323 136
394
333 338 346 322 365 349 338 349 362 301 349 385 330 384 383 384 330 301 309 354 320 368 330 365 330 341 338 303 303 360 338
349 - 4 135 60 149 56 375 - 1 6 46 195 250 28 341 0 221 36 30 252 194 35 44 194 390 -1 41 194 355 7 5 360 108 71 252 23 115 61 194 37 164 53 46 172 181 50 157 52 267 26 125 62 105 68 203 39 135 55 24 230 441 - 2 2 60 132
Density Estimates
REFERENCES A n d e r s o n , G. D. (1969). A c o m p a r i s o n of m e t h o d s f o r e s t i m a t i n g a probability density function. Unpublished doctoral d i s s e r t a t i o n , U n i v e r s i t y of W a s h i n g t o n , S e a t t l e . C e n c o v , N . N . (1962). E v a l u a t i o n of an unknown d i s t r i b u t i o n d e n s i t y f r o m o b s e r v a t i o n s . S o v i e t Math. 3, 1 5 5 9 - 1 5 6 2 . K a s s e r , I. S. and R . A. B r u c e (1969). C o m p a r a t i v e e f f e c t s of a g i n g and c o r o n a r y h e a r t d i s e a s e on s u b m a x i m a l and m a x i m a l e x e r c i s e . C i r c u l a t i o n 39, 7 5 9 - 7 7 4 . K r o n m a l , R. A. , L. B e n d e r and J. M o r t e n s e n (1970). A c o n v e r s a t i o n a l s t a t i s t i c a l s y s t e m f o r m e d i c a l r e c o r d s . J. R o y a l S t a t i s t . S o c . 19, 8 2 - 9 2 . K r o n m a l , R. A. and M. T a r t e r (1968). The e s t i m a t i o n of p r o b a b i l i t y d e n s i t i e s and c u m u l a t i v e s by F o u r i e r s e r i e s m e t h o d s . J. A m e r . S t a t i s t . A s s o c . 6 3 , 9 2 5 - 9 5 2 . T a r t e r , M. E . , R . L . H o l c o m b and R . A. K r o n m a l (1967). A d e s c r i p t i o n of n e w c o m p u t e r m e t h o d s f o r e s t i m a t i n g the p o p u l a t i o n d e n s i t y . In P r o c e e d i n g s , A s s o c i a t i o n f o r C o m p u t i n g M a c h i n e r y 2 2 , 5 1 1 - 5 1 9 . W a s h i n g t o n , D. C . , T h o m p s o n B o o k Company. T a r t e r , M . E . and R . A . K r o n m a l (1970). On m u l t i v a r i a t e d e n s i t y e s t i m a t e s b a s e d o n o r t h o g o n a l e x p a n s i o n s . Ann. M a t h . S t a t i s t . 4 1 , N o . 2, 7 1 8 - 7 2 2 . Z y g m u n d , A. (1959). T r i g o n o m e t r i c S e r i e s , V o l . II. Cambridge P r e s s .
395
London,
Kronmal and Tarter, discussion
DISCUSSION O F T H E K R O N M A L AND T A R T E R C H A P T E R
P.M.
Allen
I a g r e e w i t h t h e o r i g i n a l p r e m i s e t h a t if d a t a a r e g e n e r a t e d b y o b s e r v i n g continuous r a n d o m v a r i a b l e s we ought to s u m m a r i z e t h e d a t a u s i n g a c o n t i n u o u s e s t i m a t e d d e n s i t y f u n c t i o n and a c o n t i n u o u s c u m u l a t i v e d i s t r i b u t i o n f u n c t i o n . H o w e v e r , I t e n d to t h i n k t h a t an a r t i s t w o u l d do a good j o b of d r a w i n g a d i s t r i b u t i o n f u n c t i o n . If t h e a r t i s t d r a w s h i s c u r v e t h r o u g h t h e c e n t e r of the " j u m p s " of t h e e m p i r i c a l d i s t r i b u t i o n f u n c t i o n , t h e c u r v e w i l l be a good c o n t i n u o u s a p p r o x i m a t i o n of t h e e m p i r i c a l d i s t r i b u t i o n function. Density functions p r e s e n t a m o r e difficult p r o b l e m . Dr. Edwin Chen m a d e s o m e h i s t o g r a m s using the d a t a p r e s e n t e d in the K r o n m a l - T a r t e r p a p e r . S o m e of t h e s e h i s t o g r a m s w e r e m u l t i m o d a l while the corresponding polynomial approximations were u n i m o d a l . T h e l e n g t h and l o c a t i o n of t h e i n t e r v a l s h a s l o n g b e e n a p r o b l e m w h e n m a k i n g h i s t o g r a m s . It a p p e a r s t h a t t h e c h o i c e of t h e d e g r e e of t h e a p p r o x i m a t i n g p o l y n o m i a l is a l s o a d i f f i c u l t p r o b l e m . I h a v e b e e n l i s t e n i n g to D r . T u k e y ' s c o m m e n t s at the e n d of e a c h p a p e r . M y i n t e r p r e t a t i o n of e a c h of h i s c r i t i q u e s h a s b e e n " h a n g l o o s e and t r y s o m e t h i n g e l s e . " T h e s o m e t h i n g e l s e I s u g g e s t i n t h i s c a s e i s to p r e s e n t a s u c c e s s i o n of e s t i m a t e d d e n s i t i e s . S t a r t with s o m e t h i n g that is as flexible as p o s s i b l e , then a p p l y s u c c e s s i v e d e g r e e s of s m o o t h i n g . One w o u l d s t i l l h a v e to c h o o s e one of t h e a p p r o x i m a t i o n s and I h e s i t a t e to s u g g e s t a c r i t e r i o n f o r t h i s p u r p o s e . H o w e v e r , I t h i n k one could be g u i d e d b y t h e e x t e n t of t h e d i f f e r e n c e b e t w e e n s u c c e s s i v e d e n s i t i e s . I f e e l t h a t the t e c h n i q u e of p o l y n o m i a l a p p r o x i m a t i o n of d e n s i t i e s h a s p o t e n t i a l and t h a t a d d i t i o n a l e f f o r t in t h i s a r e a i s w o r t h w h i l e . One of t h e f o f m u l a s in t h e p a p e r a p p e a r e d t o b e s i m i l a r to t h e c h a r a c t e r i s t i c f u n c t i o n of t h e e m p i r i c a l d i s t r i b u t i o n f u n c t i o n . I w o u l d b e i n t e r e s t e d to k n o w if t h e r e i s a r e l a t i o n s h i p b e t w e e n them.
396
Density Estimates, discussion
An e m p i r i c a l B a y e s i a n m i g h t f i n d t h e s e t e c h n i q u e s u s e f u l s i n c e a n a n a l y t i c r e p r e s e n t a t i o n of t h e d e n s i t y i s g i v e n w i t h m i n i m a l assumptions.
J.
Tukey
T h e i s s u e s h e r e s e e m to m e to be i s s u e s of how w e a s s e s s b e l i e v a b i l i t y , h o w m u c h w i l l t h e g l o b a l o v e r r i d e t h e l o c a l , and h o w we d e a l w i t h t h e n e g a t i v e d e n s i t i e s w h i c h w i l l i n e v i t a b l y o c c u r f o r s o m e s e t s of d a t a . T h e f i r s t and t h e t h i r d l e a d u s to w a n t to l o o k at \[£ r a t h e r t h a n f. B o t h h a v e t e n d e n c i e s to b r i n g i n c o m p a r i s o n d i s t r i b u t i o n s . T h e s e c o n d l e a d s u s t o look at t h e actual smoothing kernel. I f e e l t h a t t h e s i m p l e s t and m o s t e f f e c t i v e a p p r o a c h to t h e f i r s t issue is that associated with hanging or suspended r o o t o g r a m s . T h e b a s i c i d e a i s s i m p l e . F i t s o m e s o r t of c o m p a r i s o n d i s t r i b u tion. Divide the line or plane up into c e l l s . F o r each c e l l find t h e v a l u e of 2 ( \/observed - \fitted ) o r , if w e f e e l p e r s n i c k e t y , t h e v a l u e of + 4 • observed -
1 + 4 • fitted
and d i s p l a y t h e s e v a l u e s , w h i c h , e x c e p t f o r a s m a l l a l l o w a n c e f o r t h e f i t t i n g of c o n s t a n t s t h a t w e a r e u s u a l l y w i s e t o n e g l e c t , a r e m o d e r a t e l y G a u s s i a n (0, 1). T h e a p p e a r a n c e of t r e n d s t h a t do n o t c o r r e s p o n d to m i s f i t t i n g c a n t h e n b e t a k e n a s s p e a k i n g f o r t h e e x i s t e n c e of c o r r e s p o n d i n g d e v i a t i o n s f r o m f i t . T h e c o m b i n a t i o n of s u c h a p l o t , u s i n g a G a u s s i a n r e f e r e n c e , and f i g u r e 12 w o u l d to m e o f f e r a v e r y m u c h c l e a r e r p i c t u r e of h o w s e r i o u s l y I w a s to t a k e b i m o d a l i t y (and t h e h u m p n e a r 198. 91) t h a n d o e s f i g u r e 12 a l o n e .
397
Kronmal and Tarter, discussion
T h i s h u m p ( n e a r 198. 91) c a n f o c u s our a t t e n t i o n on the p o s s i b i l i t y of the g l o b a l o v e r r i d i n g the l o c a l . While it i s p r e s u m a b l y at l e a s t quite d i f f i c u l t f o r s u c h t h i n g s to h a p p e n , what do w e know a b o u t the p o s s i b i l i t y that j u s t w h e r e the o t h e r d a t a v a l u e s f a l l h a s produced this h u m p ? T h e c o n t r o l l i n g entity i s the f u n c t i o n -2Ttik'x 2TT i k ' y (y, x ) = 2 e e = ( y - x ) k w h i c h r e l a t e s the a p p e a r a n c e of an o b s e r v a t i o n at x to n t i m e s the e s t i m a t e d d e n s i t y at y. With a l i m i t a t i o n on the s u m m a t i o n , s u c h f u n c t i o n s do h a v e r i p p l e s n e a r t h e i r c u t o f f , which c a n o f t e n be r e d u c e d b y r e p l a c i n g the " z e r o - o r - o n e " n a t u r e of the s u m m a t i o n , that i s , b y i n s e r t i n g i n t e r m e d i a t e w e i g h t s n e a r the c u t o f f . T h e f a c t t h a t the f i t t i n g p r o c e s s i s e x p r e s s i b l e a s
f(y) = ~
n
s
thus e l i m i n a t i n g any e x c r u t i a t i n g l y s m a l l v a l u e s f o r l/\fi
.
W h e t h e r the l a t t e r p r o c e s s is helpful enough f o r i = 1 o r beyond, helpful enough to c o u n t e r b a l a n c e its o v e r r i d i n g of v e r y s m a l l N/L w h e r e t h e s e a r e d e s e r v e d , i s u n c l e a r .
400
'
Citation Index
CITATION INDEX A l k e r , H . R . ( 1 9 6 4 ) . Dimensions of conflict in the G e n e r a l Assembly. A m e r . Pol. Sci. Rev. 58, 6 4 2 - 6 5 7 . Allen, D. M. ( 1 9 7 1 ) . The p r e d i c t i o n s u m of s q u a r e s as a c r i t e r i o n for selecting p r e d i c t o r v a r i a b l e s . T e c h n i c a l R e p o r t No. 2 3 , D e p a r t m e n t of S t a t i s t i c s , U n i v e r s i t y of Kentucky. A m o r o c h o , J . and A. B r a n d s t e t t e r ( 1 9 6 7 ) . The r e p r e s e n t a t i o n of s t o r m p r e c i p i t a t i o n fields n e a r ground l e v e l . J . Geophys. R e s . 72, 1 1 4 5 . A n d e r s o n , G . D . ( 1 9 6 9 ) . A c o m p a r i s o n of m e t h o d s for e s t i m a t i n g a probability density function. Unpublished d o c t o r a l d i s s e r t a tion, U n i v e r s i t y of Washington, S e a t t l e . A n d e r s o n , R . L . , D. M. Allen and F . B . Cady (1972). Selection of p r e d i c t o r v a r i a b l e s in l i n e a r multiple r e g r e s s i o n . In S t a t i s t i c a l P a p e r s in Honor of G e o r g e W. S n e d e c o r , T . A . B a n c r o f t , E d . , Iowa State U n i v e r s i t y P r e s s . A n d r e w s , D . F . , P . J . B i c k e l , F . R . H a m p e l , P . J . Huber, W . H . R o g e r s and J . W . Tukey ( 1 9 7 2 ) . Robust E s t i m a t e s of L o c a t i o n : S u r v e y and A d v a n c e s , P r i n c e t o n , New J e r s e y , P r i n c e t o n University P r e s s . A r m i t a g e , P . , C . K . M c P h e r s o n and J . B . Copas ( 1 9 6 9 ) . t i c a l studies of p r o g n o s i s in advanced b r e a s t c a n c e r . J . Chron. P i s . 2 2 , 3 4 3 - 3 6 0 .
Statis-
Atkins, H. , R . D . Bulbrook, M. A. F a l c o n e r , J . L . H a y w a r d , K . S . M a c l e a n and P . H . S c h u r r ( 1 9 6 4 ) . U r i n a r y s t e r o i d e s t i m a t i o n s in the p r e d i c t i o n of r e s p o n s e to a d r e n a l e c t o m y o r h y p o p h y s e c t o m y . L a n c e t , Nov. 2 8 , 1 1 3 3 - 1 1 3 6 .
401
Citation Index
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Reply to Schwartz's c o m -
B i n g h a m , C. , M . D . G o d f r e y and J . W . T u k e y ( 1 9 6 7 ) . M o d e r n t e c h n i q u e s of p o w e r s p e c t r u m e s t i m a t i o n . I. E . E . E. T r a n s a c t i o n s o n Audio and E l e c t r o a c o u s t i c s , A U 1 5 , N o . 2, 5 6 - 6 6 . B r i e r , G. W. and R . A. A l l e n ( 1 9 5 1 ) . V e r i f i c a t i o n of w e a t h e r f o r e c a s t s . In C o m p e n d i u m of M e t e o r o l o g y , T . F . M a l o n e , Ed. , B o s t o n , A m e r . M e t e o r o l . Soc. , 8 4 1 - 8 4 8 . B u l b r o o k , R . D . , F . C. G r e e n w o o d and J. L . H a y w a r d ( I 9 6 0 ) . S e l e c t i o n of b r e a s t c a n c e r p a t i e n t s f o r a d r e n a l e c t o m y o r h y p o p h y s e c t o m y b y d e t e r m i n a t i o n of u r i n a r y 1 7 - h y d r o x y c o r t i c o s t e r o i d and a e t i o c h o l a n o l o n e . L a n c e t 1, 1 1 5 4 . C a c o u l l o s , T. ( 1 9 6 6 ) . E s t i m a t i o n of a m u l t i v a r i a t e d e n s i t y . Ann. I n s t . S t a t i s t . M a t h . T o k y o 18, 1 7 9 - 1 8 9 .
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